A goal from Jack Marriott saw the Shrimps slip to an eighth defeat in nine games.
Marriott scored the only goal of the game with 13 minutes remaining with the Shrimps left wondering how they didn't get anything from the game.
Without boss Jim Bentley due to illness, a solid performance will be of little consolation to The Shrimps who are now nine games without a win.
It was a low key opening at Kenilworth Road as the hosts played some decent football without testing Barry Roche.
For The Shrimps Paul Mullin and Kevin Ellison saw tame shots saved in what was a poor spectacle.
Midway through the half Sheehan had a half chance for Luton but saw a shot with the outside of his left foot go a few yards wide.
After a shaky start The Shrimps grew into the game with Aaron Wildig seeing a deflected shot go just wide after a flowing move involving Jamie Devitt on the half hour.
Ellison had The Shrimps’ best chance of the half 10 minutes before the break but after turning well in the area scuffed his effort with Mark Tyler saving comfortably.
Luton should have gone in front on 42 minutes but after Marriott had broken in behind Goodall his low shot across the face of goal went narrowly wide of the far post.
The Hatters were the first to go close after the break, but Roche did well to deny Craig Mackail-Smith after he had been found in behind by Cameron McGeehan.
McGeehan then had a chance of his own on 53 minutes but after cutting in from the left he saw his right-footed shot deflected wide.
Again however it was a half where both sides were a good final ball short of some decent chances.
Mullin did see a tame free-kick saved by Tyler 25 minutes from time and Stephen O’Donnell brought a near post save from Roche five minutes later but shots on target were a rare sight.
With the game seemingly drifting the hosts went in front 13 minutes from time.
McCourt drove at the Shrimps defence and fed a delicate ball in behind for Marriott who slid the ball under Roche from a narrow angle.
Morecambe nearly hit straight back but the impressive Chris Doyle saw a header saved by Tyler with 10 minutes to play.
There was no onslaught from Ken McKenna’s side though, McCourt bringing a good save from Roche at the other end three minutes from time.
The Shrimps did have a couple of good chances in stoppage time but Tom Barkhuizen could only lift the ball over the bar from a couple of yards out after the returning Luke Conlan had fired goalwards before Ryan Edwards put a free header over the bar.
Luton Town: Tyler, O’Donnell, Smith, Cuthbert (capt), McGeehan, Marriott, McCourt (Mpanzu 90), Lee, Mackail-Smith, Howells, Sheehan. Subs not used: Justham, Lawless, Benson, Green, Okuonghae, Pigott.
Morecambe: Roche, Doyle, Edwards, Goodall (Molyneux 82), Beeley (capt) (Conlan 42), Kenyon, Wildig, Barkhuizen, Devitt, Mullin, Ellison. Subs not used: O’Hara, McGowan, Wilson, Hedley.
Referee: Darren Deadman
Attendance: 7,153 (56 Morecambe)