Keith Keane's own goal in the fifth minute set the Swans on their way, before Guillem Bauza made it 2-0 in the 64th minute with his first English league goal.
Paul Furlong's tenth of the season, with his first touch just seconds after coming on as a 74th minute substitute, briefly threatened a comeback.
But Jason Scotland sealed the points with a thunderous finish eight minutes from time.
With skipper Chris Coyne and Wales international Dave Edwards sold in midweek and Matthew Spring suspended, Luton made three changes from the side that bravely held Liverpool 1-1 in the FA Cup last Sunday.
In came Don Hutchison, Richard Jackson and Steve Robinson for the hosts, while the visitors made four changes from their last outing.
If things had been difficult off the pitch for the Hatters, they didn't get any easier on the field as Swansea opened the scoring just five minutes in.
Andy Robinson's cross-shot from the left took a deflection on its way into the penalty area and with Dean Brill beaten, the ball appeared to take a nick off the unfortunate Keane on its way into the far corner of the net.
Swansea came again in the 15th minute as Scotland found himself clean through in the penalty area only for Brill to produce a superb block.
Brill appeared to injure himself in the process of making the save, but with no substitute goalkeeper on the bench he was forced to soldier on.
The half ended with Drew Talbot nodding Darren Currie's header straight at Dorus de Vries in stoppage time.
Luton were hit with a further setback in the second period when stand-in captain Chris Perry was forced to limp off in the 57th minute with Lewis Emanuel stepping in.
And the visitors exposed the hosts' weakened back line with a second goal in the 64th minute.
Breaking from a Hatters corner, Angel Rangel fed Darren Pratley breaking down the right and the midfielder in turn played in Bauza to prod the ball past Brill from just inside the penalty area.
The home side made a change, taking off the wretched RJackson for veteran striker Furlong in the 74th minute and the ex-Chelsea and QPR man made an immediate impact scoring with his first touch from a yard out after Talbot's shot from Emanuel's cross had hit the post.
But any hopes of a home comeback were dashed when Scotland rifled home a third from 20 yards with eight minutes remaining.