The Hatters will travel to Grimsby for a replay after being held to a 2-2 draw by the Sky Bet League Two side at Kenilworth Road.
The Mariners took a first-half lead through Gavan Holohan's header, but Elijah Adebayo's penalty and Jordan Clark's volley put the Town in front midway through the second half, only for Grimsby's Harry Clifton to net an immediate equaliser to forced a deserved rematch in Cleethorpes on Tuesday week.
Manager Rob Edwards handed Pelly-Ruddock Mpanzu his 350th Luton Town appearance, with the midfielder - who signed permanently for the club on this day in 2014 - starting alongside Jordan Clark and Allan Campbell as the boss named a strong line-up for the visit of the Mariners.
Gabe Osho started for the third match in succession after being forced off against Wigan Athletic last weekend, whilst Dan Potts captained the side with Tom Lockyer in-between the pair.
Seventeen-year-old Zack Nelson was among the nine substitutes after impressing during the FA Youth Cup victory against QPR U18s ten days ago, while new signing Cody Drameh was ineligible for today's fixture having played for Leeds United in the competition already this season.
After the sprinklers had come on unexpectedly within seconds of the match kicking off, causing an early break, it was the visitors who had most of the territory in the opening exchanges, although Mpanzu had the first sight of goal, hammering an attempted volley into the ground and wide.
The Mariners survived a penalty appeal when Adebayo's cross from the left hit Gavan Holohan's right hand in the box, and Potts had a header ruled out for offside with 11 minutes on the clock.
Adebayo and Cornick linked up well just before the 20-minute mark, playing a one-two on the edge of the box before the former fired over the bar. Then Bell delivered a deep cross from the left that Adebayo tried to volley back across the face of goal, only for Crocombe to block behind.
Mpanzu hit a first-time 25-yarder well over the bar on the half-hour, before Bryn Morris had the game's first shot on target at the other end, testing Horvath with a drive from outside the box, but the USA international was equal to it.
The Hatters continued to probe, a succession of corners leading to Clark tricking his way past a defender - and not for the first time, after an earlier bit of ball juggling on the halfway line as he brought it forward in full flow - to volley a cross in that ended up landing invitingly for Cornick, who couldn't keep his volley down.
It was Horvath who was called into action again a couple of minutes before half-time, as Holohan nipped in front of Osho as the Town defender hesitated and hit a dipping volley from 30 yards that the Town keeper had to tip over his bar.
From the resulting corner, the Mariners' midfielder climbed highest through a pack of players and planted a free header into the corner of the net to give the visitors a half-time lead.
That advantage was wiped out within five minutes of the restart, however, as Osho had his legs taken away from under him as he beat Anthony Glennon just inside the area. Referee Tim Robinson pointed to the spot and Adebayo did the rest, sending Crocombe the wrong way as he netted his seventh goal of the season.
Grimsby's Alex Hunt curled a 20-yard free-kick over the bar in the 53rd minute, but Town were soon back on the attack, Adebayo driving towards the left byline and
Edwards made his first changes on the hour, replacing Campbell and Cornick with Cauley Woodrow and top scorer Carlton Morris.
The reward came in the 66th minute with Clark receiving Doughty's low cross around the penalty spot, flicking it up and volleying right-footed into the top corner, with Crocombe getting a hand to the ball but unable to keep it out.
The visitors were back on terms within a minute, as Clifton raced into the Town box and finished unerringly across Horvath and into the far corner of the net.
Crocombe somehow managed to claw a Morris header away as the Town striker met Ckark's lofted cross at the near post, before Holohan again created space for himself just outside the D at the other end, although this one was straight at Horvath.
Having already brought Onyedinma on for Potts, Edwards introduced Luke Berry in place of Mpanzu with eight minutes to go, but it was Osho who had to react smartly to clear Clifton's effort from almost on the line as the Mariners once again asked the questions.
Berry and Woodrow both stood inviting crosses up into the Grimsby six-yard box as the Hatters pushed for a winner, while Berry volleyed Doughty's deep cross at Crocombe before disappearing head first over the advertising boards and into the crowd.
But ultimately, neither side could make the crucial third breakthrough in seven minutes of time added on and will both enter tomorrow's fifth round draw as ball number 6, before renewing acquaintances at Blundell Park.
LT - Adebayo (pen) 49, Clark 66
GT - Holohan 43, Clifton 67
Att: 8,552 (1,473 away)