A solitary goal from Tom Craddock - his fourth in three games - earned the Hatters' their third straight victory and moved them to within five points of the top of the Blue Square Premier.
The striker's 36th-minute effort, together with a solid defensive display, was enough to ensure the Town's first clean home clean sheet in the league since a 1-0 win over Barrow on 12th September last year.
The Hatters enjoyed a bright start and dominated a first half playing some bright, attacking football on a pitch still covered in plenty of sand.
Keith Keane came close to giving the hosts the lead as early as the fourth minute when he saw a shot saved by Sam Russell in the visitors' goal.
However, despite their dominance, Russell had barely been extended but as the half wore on the Town upped the tempo and deservedly took the lead nine minutes before the break.
Good defending down the Town left eventually saw the ball moved forward into the path of Craddock.
The striker twisted one way and then the next to get past the Red Dragons' giant centre-back Mansour Assoumani before firing a fierce left-foot shot towards the top-corner that was tipped onto the post by Russell before the ball&nbsp;spun over the line.
The opening goal&nbsp;was no more than the Town deserved with Keane in particular running the show in the centre of midfield alongside tenacious debutant Simon Heslop.
Two minutes after Craddock's strike Claude Gnakpa came close to making it 2-0 with a header straight at Russell after Jake Howells' nimble footwork allowed him to centre from the left.
Wrexham fought back well though, and &pound;30,000 signing Andy Mangan - the BSP's top goalscorer last season - should have done better when well-placed on 40 minutes&nbsp;to meet Luke Holden's free-kick with Janos Kovacs in the way to block.
The Town started the second half&nbsp;the way they started the first&nbsp;and, after Keane had got the crowd in stitches after accidentally kicking the corner flag instead of the ball, his second attempt led to Howells drilling a shot inches over the crossbar on 52 minutes.
After that escape, Dean Saunders' side got a grip on proceedings and came within a whisker of finding an equaliser on 68 minutes.
The Hatters failed to clear a right-wing corner that eventually found impressive midfielder Andy Fleming who let from the left-hand edge of the penalty area.
With the ball rocketing goalwards, thankfully for the Town, goalkeeper Mark Tyler was able to get the faintest of fingernails onto the ball to divert the shot onto the crossbar to keep the scores at 1-0.
Still the Dragons pushed in search of a leveller, pushing the&nbsp;Town back, but Tyler was rarely tested again.
And despite some nervy late aerial assaults from the visitors, the Hatters held on comfortably enough to rack up win number three on the spin to keep them third in the table.
With Oxford drawing with Kettering and Stevenage in FA Trophy action the Town cut the gap at the top to five points.
Who knows what will happen, but make sure you get along to Kenilworth Road on Tuesday when Kidderminster are in Town as we chase a fourth straight win.
For minute-by-minute analysis, see below.
Result from Kenilworth Road
Luton Town 1 (Craddock 36)
FULL-TIME LUTON TOWN 1-0 WREXHAM - Another victory...all the reaction to come. Get in!
90 + 4 mins - 30 seconds left...
90 + 2 mins - All Wrexham now as Tyler does brilliantly to prevent Murray's backward header from going behind for a corner. Three minutes left now.
90 mins - We've got five additional minutes to play for the Hatters to hold on. It does seem a little too much.
89 mins - All hands to the pump again for the Town as Tyler punches clear from a Wrexham corner as the visitors press. The Nwokeji races clear but he has no support and the move breaks down.
87 mins &nbsp;Second change for the day for the Hatters as Nwokeji replaces the goalscorer Craddock. This is the substitutes' first outing at Kenilworth Road since joining in November. Meanwhile, Keane is named the sponsors' man-of-the-match. Deservedly so.
86 mins - Half a chance for the Town as Jarvis centres towards Gallen with a lovely right-wing cross that eludes Gallen. It looks as though the ball will drop to Howels as the back post but Assoumani is on hand to clear the danger by sticking a leg out at the last minute.
85 mins - Not much to tell you about in all honest and that could be said for much of this second half. Five minutes left...
81 mins - Two half chances in the space of 30 seconds for the hosts as firstly Murray hits a shot goalwards that hits Gallen before Heslop shoots over the crossbar.
80 mins - The Kenilworth Road are getting behind the Hatters as they go in search of a second - can they get one? Elswhere in tne BSP Rushden are beating Gateshead 7-0.
76 mins - Very frustrating pout there at the minute as the Town search for a second&nbsp;but need to be aware at the back. The referee hasn't won any friends in the home crowd with some strange decisions as Howells goes sprawling under Spann's hefty challenge. Still 1-0...
71 mins - All Wrexham now - the Town need to ride this storm. All hands to the pump.
68 mins - Almost one a piece. From the resulting corner, the Town fail to clear their lines. The ball falls to Fleming on the right-hand edge of the penalty area who unleashes a super right-foot shot that crashes off the underside of the crossbar. Tyler may have got a finger on that - if so that was absolutely crucial. We'll have to ask him afterwards. That was very, very close. The Hatters desperately need a second.
67 mins - Pressure from Wrexham as they force their first corner as Sakho robs Asafu-Adjaye of the ball inside the penalty and squares for Mangan who sees his shot blocked by Murray. Better from the visitors.
66 mins - The attendance at Kenilworth Road is 6,528 which includes 143 shouting for the Welshmen.
64 mins &nbsp;First change from Richard Money as Claude is replaced by Rossi Jarvis. There's activity on the opposition bench too as Obeng comes on for Jones.
62 mins - With Spann lying lame in the away penalty area the Town decide to play on and Craddock should do better when unmarked on the left hand side of the penalty area.
61 mins - Almost half a chance for Craddock sniffing for his second as he latches onto Tyler's long boot down field but Russell, the Wrexham keeper, does well to react first to clear the danger.
60 mins - It's not been the most enthralling first 15 minutes of football, it's been pretty even-stevens. Wrexham have improved slightly as Mangan tries a shot from outside the penalty area that is comfortable for Tyler. Still 1-0.
54 mins - First change of the afternoon is from Dean Saunders as Sakho replaces Holden.
53 mins - The ref really should have booked the corner flag for that foul on Keane. That was quite clearly deliberate...
52 mins - Comic moment from Keano as he attempts a right-wing corner he kicks the corner flag instead of the ball! Take two sees the midfielder find Heslop who squares for Howells who steadies himself before firing a shot narrowly over Russell's crossbar. Unlucky.
50 mins - The Hatters do need a second as they know Wrexham will attack in search of an equaliser. They break with Spann, the right-back, who whips in a teasing cross but Tyler is there to make a good catch. Easy pickings.
48 mins&nbsp;- Craddock wants a second and does well to keep the ball down the left-hand touchline. He cuts inside, beats Spann as he enters the penalty area but he can only find the gloves of Russell with a tame right-foot shot. Good work rate from the striker, though.
46 mins Wrexham get the second period underway with the Town now kicking towards the Oak Road end where around 150-odd Dragons fans are sitting. Can the Town get an early second? COYH!
HALF-TIME LUTON TOWN 1-0 WREXHAM - That's the end of a very satisfactory first 45 minutes and Craddock - who is enjoying a hot streak in front of goal at the minute - is the man who has put the Town ahead and deservely so. Check back in 15 minutes for the second half updates. COYH!
45 mins - We've got two additional minutes at the end of the half. Can we get another before the break?
42 mins - Heslop has been impressive this afternoon, he comes forward with the ball and fancies a goal himself now. Unfortunately he gets his shot all wrong and the ball drifts wide.
40 mins - Chance at the other end for an equaliser as Holden picks out Mangan with a square pass from a left-wing free-kick. The Wrexham striker hits it first time but Kovacs is there to clear the danger.
38 mins - Almost 2-0 as the Town look to put this to bed before the break. Howells centres from the left and Gnakap gets up to head straight at Russell.
36 mins &nbsp;GOAL! HATTERS 1-0 Wrexham - The answer to my question on 35 minutes is yes! Craddock runs down the left flank, cuts inside, beats Assoumani and fires a vicious left-foot shot that Russell turns onto the post but the momentum of the ball takes the ball over the line for the striker's 14th goal of the season and his fourth goal in three and a bit games. Get in!
35 mins - The Hatters are turning the screw but can they find a goal? More super football involving Heslop, Craddock and Gallen sees the ball pinged out to Claude down the right and his firm cross eludes everyone and Wrexham are able to clear.
32 mins - Looking at the other scores in the BSP - there has only been one goal and that's come for our promotion rivals Rushden at home to Gateshead. Can we have a goal here (for us) please?
30 mins - Back come the Dragons as Holden picks out Baynes fires a shot from 20 yards straight down the throat and into the arms of Tyler after taking a lovely cross-field pass from Holden.
26 mins - More good play from the Hatters as good link-up play on the edge of the Wrexham penalty between Heslop and Craddock sees the former in on goal but Craddock contrives to get in his way and the debutant who can only shoot into the ground as the ball rolls safely into the arms of Russell.
25 mins - The Town are upping the ante now as Asafu-Adjaye, who has done well so far this half, whips in a dangerous low cross from the right aimed for the near post that Gallen is inches away from connecting with. Keane, Gnakp and Craddock were also involved in the build-up. Unlucky.
23 mins - Better from the Hatters as a short corner routine almost pays off as Keane touches it to Howells who drives forward into the box and sees a shot drift just wide of the right-hand post.
19 mins - The Town are still marginally on top but we've yet to have a real clear-cut chance yet. Heslop has impressed in the middle of the park so far and he tests Russell in the visitors' goal with a cross-cum-shot that is easy pickings for the goalkeeper, even if it did take him by surprise.
16 mins - First spell of pressure from the visitors as Christian Smith finds space inside the Town half but he can only fire a low shot wide of Tyler's left-hand post.
14 mins - Is that a half chance? Not really as Gnakpa gets involved this time to nod well, well wide following Howells' deep left-wing cross.
10 mins - Claude's the only player wearing gloves. And then he takes a foul throw...
9 mins - The Town are playing some decent stuff so far as move from the right to the left draws appreciation from the home faithful. Keane is everyone at the moment. Hounding the midfield four of Wrexham, breaking up the play and passing it well too.
8 mins - More pressure from the Hatters as Gnakpa does well to roast Williams down the right. He flashes a cross into the box that Assoumani, the giant Dragons centre-back, has to clear behind for a corner.
4 mins - Bright start from the Hatters as the visitors look nervy at the back. Keane picks the ball up 25 yards from goal and drives forward and tries a shot from just outside the box that Russell in the Wrexham goal has to get down and deal with. Positive play from the Town.
2 mins - It's a flat 4-4-2 for the Town this afternoon: Ed, Pilks, Kovacs and Murray across the back, Claude, Keano, Heslop, Gnakpa in midfield and Craddock and Gallen up top. There's plenty of noise inside Kenilworth Road early doors.
1 min It's the Hatters that get us underway kicking off towards the Kenilworth Road end. Enjoy the updates. COYH!
14:57 HERE THEY COME
Out come the two sides onto the pitch, the Town in their orange home shirts, led by George Pilkington, Wrexham in their white away shirts, red shorts and white socks.
14:51 PRESS BOX PREDICTIONS
Andrew Barringer: 3-1 Luton. Simon Pitts: 3-1 Luton. Mark Wood: 3-1 Luton. Steve Sims: 2-1 Luton. Euan Duncan: 2-1 Luton.
14:45 CLAUDE'S ODDS TUMBLE
Our advice yesterday to get on Claude Gnakpa to score either first (28/1) or at anytime (9/1) has got the bookies running for cover. Did you get on at those prices? If you haven't it's too late. Bet365 now price the Frenchman at 11/1 to score first and 100/30 at anytime. Still, not bad but not quite 28s! Let's hope he can do the business for us this afternoon.
14:40 13TH MARCH 1985
As mentioned in Roger Wash's excellent programme piece, today marks the 25th anniversary (if you can call it that) of the day that day when Millwall came to come for an FA Cup quarter final that ended with Kenilworth Road experiencing "one of the worst nights of football violence this country has ever seen".
14:15 TWO CHANGES
Town boss Richard Money makes to changes to his side that beat Forest Green 2-1 on Tuesday night.
Midfielder Simon Heslop makes his debut in the middle of midfield and Jake Howells appears on the left flank. Out go Asa Hall and Matthew Barnes-Homer - both of which don't even make the bench.
Hatters: Tyler, Asafu-Adjaye, Keane, Kovacs, Pilkington, Craddock, Howells, Gnakpa, Gallen, Murray, Heslop.
Substitutes: Blackett, Gore, Jarvis, Nelthorpe, Nwokeji.
Wrexham: Russell, Westwood, Williams, Assoumani, Jones, Spann, Baynes, Fleming, Smith, Holden, Mangan.
Substitutes: Obeng, Taylor, Sakho, Wolfenden, Maxwell.
13:35 NOT SO SANDY
The Kenilworth Road pitch was covered in 10 tonnes of sand prior to the victory over Forest Green on Tuesday. That was an improvement on what it looked like before then when it resembled a muddy bog (that saw the Histon game postponed). The sand is still there but it has clearly taken to the pitch. More sterling work from the groundstaff. It looks even better than it was on Tuesday.
12:55 HOW TIMES CHANGE
Luton Town v Wrexham. It's a fixture that most would associated as a League One or League Two match.
How times have changed.
While the Town are trying their utmost to get out of the BSP at the first attempt, Wrexham are finding their route back to the League a frustrating one.
This is their second season at this level and are marooned in mid-table going into the game today.
It seems strange given the history between the two sides that they both meet out of the Football League.
The last time the Red Dragons visited Kenilworth Road, in November 2004,&nbsp;the Hatters ran out 5-1 winners, extending their lead at the top of League 1. Remember Sol Davis scored his first goal for the club that day?
The 2004/05 season saw the Town lift the Championship by 12 points while the Welshmen were relegated to League 2.
Today, come 3pm, memories of that Hatters win will become a dim and distant memory with Richard Money's side out to win their third successive match and put pressure on the teams at the top.
Who knows how long it will be before today's game will be a League fixture again.
But let's hope we don't have long to wait...
Well, whatever happens this afternoon your official website will have everything covered: every step of the way.
If you can't make the match then we'll have all the goals and incidents as it happens live with the usual news in the run-up to kick-off with team line-ups and plenty of chat as your official website has, as ever, the usual updates keeping you bang up-to-date with all the action - and in a number of ways.
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