Christian Roberts hit the winner seven minutes into the second half to give the Robins the points in a dire clash.
The winger drilled in a low effort from ten yards to keep his side in the play-off chase and leave the Hatters deep in relegation trouble.
In front of plenty of empty seats at Kenilworth Road, the visitors got off to the better start and came close to opening the scoring in the fourth minute when Lee Peacock glanced Roberts' left-wing free-kick just past the post from ten yards.
Luton soon got to grips with their opponents and David Bell, making his return from injury, beautifully jinked past three players in midfield before smashing a 25-yard, right-footed shot just wide in the 12th minute.
The hosts had the ball in the net two minutes later via Drew Talbot, but the cheers from the home fans were cut short as the goal was disallowed.
Bell was the creator, cutting inside from the left flank away from two opponents and sending in a ferocious 20-yard drive which Petr Brezovan could only parry into the path of Talbot, who from an offside position, converted the rebound.
The two moments of skill from Bell were the best the half had to offer and the lack of quality on show was highlighted by an unbelievable miss from Moses Ashikodi after 22 minutes.
The striker did well to spin away from Keith Keane after Calvin Andrew had misjudged a ball on the edge of the area, but from eight yards out and with just Dean Brill to beat, Ashikodi blazed his shot well clear of the crossbar and into the back of the Kenilworth Stand.
Luton made two substitutions at half-time in an effort to lift the tempo as Sam Parkin replaced Andrew and Dean Morgan came on for Darren Currie.
But it was Swindon who took the lead in the 52nd minute as Roberts picked up Jon-Paul McGovern's long pass on the right flank and ghosted past half-hearted tackles from Alan Goodall and Chris Perry before firing in a low left-footed drive from ten yards.
The Robins almost made it two five minutes later as Roberts again beat Goodall with ease on the right and crossed for Peacock to see his 12-yard header blocked on the line by Richard Jackson.
A dreadful game saw Swindon comfortably deal with the little the home side had to offer as Luton struggled to put the visitors under any meaningful pressure.