Ashley Barnes' late penalty was enough to secure victory at Kenilworth Road on Saturday afternoon and extend the Clarets' unbeaten run in the Sky Bet Championship to 12 matches.
Burnley were awarded a spot-kick in the 78th minute for a handball and Barnes made no mistake from six-yards to seal all three points.
Manager Vincent Kompany made two changes to the side that drew 1-1 with Watford on Tuesday night at Turf Moor.
Charlie Taylor came into the defence in place of the injured Jordan Beyer with Jack Cork returning to the XI replacing Anass Zaroury.
First Half
The Hatters had the ball in the back of the net just five minutes in, but Morris’ strike was ruled out following a foul from Adebayo on Barnes inside the area during the corner.
Five minutes later Morris’ strike partner Adebayo had a chance of his own, after a neat one-two with Morris, the forward’s effort was pushed away for a corner by Muric.
The Clarets’ first opportunity on goal came on the half hour mark when Gudmundsson saw his goal-bound free-kick from 30-yards out palmed just wide for a corner.
Both teams were pushing for an opener before the break and Luton came close once again when Mpanzu struck a fierce effort from just outside the box but Muric had it covered.
Just two minutes later Town had another chance when Mpanzu turned provider and the midfielder picked out Adebayo unmarked inside the area only to fire well over the bar from close range.
Second Half
Kompany then turned to his bench with eight minutes of the second half played in search of an opener, as Zaroury replaced Gudmundsson out wide.
The Clarets were then dealt a blow on the hour when Tella was forced off through a knock with Vitinho coming on for the forward.
The Brazilian substitute was straight into the action and looked like he had carved open the first goal of the afternoon, only for an excellent Lockyer block to deny the drilled effort from just yards out.
Burnley were knocking on the door as the game headed into the final 20 minutes at Kenilworth Road and forced a corner when a dangerous cross from Maatsen was tipped just over the bar from Horvath.
But on the 76th minute the Clarets were awarded a spot-kick for handball after Vitinho broke into the are a saw his dazzling run denied by the arm of a Town defender.
Barnes stepped up to the spot and fired home past Horvath in the Luton goal and wheeled away in celebration in front of the travelling Clarets.
Burnley: Muric, Roberts, Taylor, Ekdal, Maatsen, Brownhill, Cullen, Cork, Gudmundsson (Zaroury 53) (Foster 90), Tella (Vitinho 60), Barnes
Unused Subs: Peacock-Farrell, Al-Dakhil, Twine, Obafemi
Luton Town: Horvarth, Drameh (Doughty 32), Burke, Lockyer, Osho, Bell, Nakamba (Campbell 68), Mpanzu, Clarke, Morris (Woodrow 57), Adebayo
Unused Subs: Shea, Potts, Berry, Taylor
Goals: Barnes 77' (Pen)
Attendance: 10,066 (1,026)