Town moved up to fifth in the Sky Bet Championship table with a second win of a three-match unbeaten week, courtesy of Morris' first-half tap-in.
Manager Rob Edwards made one change with Allan Campbell, who netted the Town's first goal in the 2-0 win Swansea back in August, selected to start a league game for the first time since the win over Stoke City a month ago.
The Scotland international lined up in midfield alongside Marvelous Nakamba and Jordan Clark, with Pelly-Ruddock Mpanzu named among the substitutes in the only switch from Tuesday night's 2-2 draw with Millwall.
The Hatters kicked off in sixth place in the table, occupying the final play-off spot, while the visitors from south Wales were in 15th position, 11 points behind Town with both clubs having played 34 matches.
Ethan Horvath was the first goalkeeper to be called into action, diving low to his left to keep out an 11th-minute shot from Luke Cundle.
But the Hatters were soon asking questions of their own, Elijah Adebayo climbing highest to meet Alfie Doughty's corner a couple of minutes later, but sending his header over the bar.
Top scorer Carlton Morris was next to have a go, Doughty and Adebayo causing problems down the left and when the former delivered another telling cross to the far post, Morris' downward header was parried by keeper Andy Fisher and gathered at the second attempt as Campbell prepared to pounce.
Town were picking the loose balls up as Swansea attempted to play out, Nakamba smothering them at every opportunity. But i it was from a longer goal-kick clipped into wide space in the centre circle that they nicked another, this time Campbell feeding Clark in the box, although Gabe Osho couldn't keep his eventual shot down.
Cundle became the first player to be shown yellow for pulling Amari'i Bell back just before the half-hour as the left-sided centre-back looked to race clear in the Swansea half. Doughty whipped in another terrific cross, but Clark's flick went begging at the far post.
Morris sent a thumping header that clipped the crossbar from a Clark corner in the 33rd minute, seconds after Swansea striker Joel Piroe had proved effective at the other end, diverting a low cross from Clark away from the former Barnsley hitman in the visitors' six-yard box.
The Welsh side were hanging on as Town piled the pressure on, Lockyer meeting another Doughty corner with a header that Jay Fulton somehow managed to chest off the line, with his goalkeeper patting the ball away from danger as the Hatters looked to capitalise.
The breakthrough finally came six minutes before half-time as the Hatters' relentlessness paid off, Doughty's initial cross from the left finding Cody Drameh, whose volley was blocked into the path of Clark, who also saw his shot diverted to Campbell.
The Scot had the presence of mind to feed Doughty once more on the left edge of the box, and when Fisher couldn't hold the ex-Stoke man's low delivery, Morris was there to snap up his 14th goal of the season.
Swansea midfielder Ollie Cooper picked up an early booking after the interval for kicking Lockyer in the face. But it was the City midfielder who then spurned a great chance to level in the 48th minute when he burst clear of his fellow Welshman and into the box, but ballooned his shot over the bar.
Clark soon followed Cooper into referee Dean Whitestone's book for a tug back on Ryan Manning in front of the Enclosure, and although Swansea had come out as a different side for the second half, it was Town who created the first real chance of the half.
Clark skipped past his man on the right edge of the box and centred for Campbell, whose glancing header at the near post was parried by Fisher, and with Morris primed to tap in again, visiting defender Ben Cabango reacted quickest to stab the ball wide of the post, although referee Whitestone was convinced it had come off the Town striker and gave a goal-kick.
Edwards made his first switch on 68 minutes, replacing Adebayo with Cauley Woodrow, while Swansea boss Russell Martin - who had already brought on Morgan Whittaker for the injured Joe Allen in first-half injury-time - switched Cundle and Matthew Sorinola for Liam Cullen and Liam Walsh.
Woodrow was the link man as Town created their next chance in the 72nd minute, Campbell winning the ball back in midfield and feeding the former Fulham and Barnsley forward to play Doughty in on the left, although the wing-back's shot was always rising into the Kenny End.
Morris fired into the side-netting with 15 minutes to go after Woodrow flicked on and Clark slipped the ball into the striker's path to strike at goal from 20 yards.
Lockyer and Doughty combined to set Woodrow up for another shot with just over ten minutes remaining, with ref Whitestone waving away appeals for a penalty with Nathan Wood blocking with his arm, but tightly tucked into his body.
Morris was taken off to a standing ovation, replaced by Pelly-Ruddock Mpanzu, but it was Swans sub Whittaker who was soon reminding Horvath that the Welshmen still carried a threat, the American beating his 86th-minute shot away.
Clark hit the outside of Fisher's right post with a clever 'Don't Watch the Eyes' sidefoot shot on 87 minutes, after Doughty had played man-of-the-match Drameh's diagnonal ball into his path.
City boss Russell Martin and Town assistant Richie Kyle were sent off for their park in a sideline melee caused after a foul on Nakamba was missed by ref Whitestone, and Town saw out the added time to secure a tremendous seven point week.
Goals:
Morris 39'
Att: 10,072 (1,031 away)
