Morris headed home Alfie Doughty's cross in the 47th minute to secure maximum points for the Town, and earn a first win in four matches, although the game was soured by the top scorer having to leave the action through injury in the second half.
Recent performances had merited more, and manager Rob Edwards made one change with the suspended Tom Lockyer replaced by Alfie Doughty, who returned to the starting line-up after coming off the bench against Burnley last weekend.
Fred Onyedinma replaced the number 45 and returns to the substitutes' bench having missed out against the Clarets
Midfielder Pelly-Ruddock Mpanzu captained the side for the second time, the first being in the 1-0 win at Coventry last March, as the Hatters looked to return to winning ways after three league games without taking maxiumum points.
Amari'i Bell made a brave block from Tahith Chong's early shot, but Town settled quickly and were playing the better football when Jordan Clark did well on the right to set the ball back for Cody Drameh to deliver a cross that found Elijah Adebayo's head, but the striker couldn't find the target.
Clark then had a tame shot that was comfortable for John Ruddy to gather, before Scott Hogan headed Lukas Jutkiewicz's cross well wide for Birmingham in the 23rd minute, shortly after Doughty had become the first player into referee David Webb's book for a trip on Chong.
The Hatters' flying left wing-back was soon making inroads at the other end of the pitch, nutmegging his way past Maxime Colin before delivering an inviting cross for Adebayo, who headed over at the far post.
Town were creating all the decent openings in the game, with Bell switching the play for Drameh to deliver a low cross - after trading passes with Adebayo - for Clark to strike at goal, although for once the midfielder miscued from 12 yards out.
The breakthrough came two minutes into the second half, and just after Clark had danced his way to the byline and cut a ball back that Birmingham managed to clear, it was top scorer Morris who made it 13 for the season, meeting another wonderful cross from Doughty with a downward header that bounced up and past Ruddy into the top corner.
Clark tested Ruddy again with a shot from just outside the area, before the Hatters had to reshuffle the pack with Burke picking up a knock that forced his substitution on 56 minutes, Allan campbell coming on in his place with Mpanzu dropping back to right-sided centre-half.
Morris was soon step-overing his way to the left byline and zipping a ball across the six-yard box that just eluded Campbell and Adebayo, with George Hall grateful to clear at the far post.
Mpanzu might have been moved back, but the skipper for the day delivered another delicious cross in the 62nd minute that once again found Morris climbing highest, only for his glancing header to drift just wide of the far post.
Birmingham made a triple change on 66 minutes, and one of the replacements - Marc Roberts - was soon the centre of attention for a terrible challenge on Morris, with both feet off the ground, that eventually resulted in the Town striker's early exit through injury.
Somehow referee Webb only saw fit to show the centre-half the same colour card that he had Doughty for his first offence, a tame trip on Chong in the first half.
Roberts was almost ibn the right place to profit from a goalmouth scramble on 77 minutes, with Kevin Long heading Juninho Bacuna's free-kick back across goal and Mpanzu, Bell and Campbell all playing a part in blocking the Blues' path before Drameh led the ball away from Roberts at the far post to avert the danger.
The hosts came closer than ever to levelling in the 79th minute when Hogan headed Chong's right-wing cross against the bar, but Town's makeshift defence - a midfielder, a left-back and a centre-half who also plays midfield between them - stood firm with Drameh and Doughty excellent alongside them.
Substitute Cauley Woodrow hit a powerful 25 yarder that Ruddy almost fumbled over his line with 90 minutes up, but the Hatters had done enough to see out another big win to keep them inside the play-off places ahead of Tuesday's live Sky game against fifth-place Millwall.
Goals:
Morris 47
Att: 16,866 (1,838 Hatters)