Kevin Gallen was the Hatters' hero with a 59th-minute strike to give the visitors a vital victory against the in-form Shrimps who were unbeaten for 12 games before this encounter.
Morecambe dominated the early stages but Luton looked sharp on the break and it was the visitors' Tom Craddock who had the first clear chance, forcing Barry Roche into a low save with a 15-yard shot on 18 minutes.
But Morecambe went straight up the other end and took the lead on 21 minutes when Stewart Drummond rose superbly at the far post to head home a pinpoint left-footed free-kick from Michael Twiss.
Twiss had a great chance to create a second a minute later but let the ball run out of play when he had sprinted clear of his marker to chase into the box.
The Hatters thought they had equalised after 24 minutes but the referee signalled there had been a push before midfielder Asa Hall had headed a Kevin Nicholls cross past Roche.
Despite the disappointment, the visitors stepped up a gear and, after a period of dominance got their reward in the 43rd minute, when Chris Martin got in front of Dave Artell to head a Nicholls cross past Roche.
And they almost took the lead just before half-time when Martin squeezed a shot under Roche only to see Drummond clear the ball off the line.
Luton started the second half strongly with the ever-dangerous Craddock forcing Roche to make a superb save after a weaving run into the box which ended with Garry Hunter reacting sharply to deny Martin from stroking home the rebound.
But Luton did go ahead in the 59th minute when veteran striker Gallen poked the ball home from what looked suspiciously like an offside position as Morecambe failed to clear a hopeful ball into the box.
Morecambe responded well and Twiss and Hunter both forced saves from Dean Brill with shots from acute angles while Curtis flashed a shot over and debutant Mark Duffy fired into the side-netting after a good run.