Andre Ayew grabbed his 10th goal of the season to earn the Swans a deserved victory at Kenilworth Road.
With the game heading towards a goalless draw, Bersant Celina and Jay Fulton combined superbly to set up Ayew with a simple finish nine minutes from time.
Victory moved the Swans into the play-off spots at the halfway stage of the season.
The Swans made three changes to the side that beat Middlesbrough 3-1 last Saturday.
Swans boss Steve Cooper welcomed back Ben Wilmot in the heart of defence with Ben Cabango ruled out with a slight groin issue.
Tom Carroll returned from his one-match suspension to make his 100th appearance for the club in place of Yan Dhanda, while Borja started up front instead of Sam Surridge who had a slight hamstring problem.
Hatters boss Graeme Jones, a former assistant manager with the Swans under Roberto Martinez, made three changes as his side went in search of their third consecutive home win.
Simon Sluga started in goal for the first time since October, with James Shea suffering the effects of concussion. Andrew Shinnie and Harry Cornick returned in place of Callum McManaman and Kazenga LuaLua respectively.
Both sides made a lively start to the game as George Byers forced a comfortable save from Sluga after just three minutes.
After a good spell of possession for the visitors, Luton hit back down Swansea’s right and had two penalty appeals turned down in quick succession for handball against Conner Roberts and then Byers with 11 minutes on the clock.
Swansea were also enjoying their best spells down the left despite the tight Kenilworth pitch as Bersant Celina, the Swans’ best outlet, just failed to pick out Andre Ayew with a dangerous cross.
Ayew, looking lively, fired into the side netting after 23 minutes as the Swans switched play to the opposite flank
The visitors were playing some neat football in and around the Luton penalty area as they continued to just edge the possession stakes, without testing Sluga in goal.
The first real save of the game came at the other end though as Freddie Woodman was forced to tip a fierce effort from Andrew Shinnie over the bar in the 40th minute.
Jake Bidwell and James Collins both found themselves in the referee’s notebook for fouls before Borja sent an overhead kick just wide and the half-time whistle ended an encouraging opening 45 for the Swans.
Neither side made any changes for the start of the second half as the Swans started on the front foot with Celina, Ayew and Borja combining well to force a corner which Luton cleared.
The Swans had to remain resolute though as a good forward run from Luke Bolton saw his cross find Harry Cornick unmarked at the far post, but his shot flew wide.
Tom Carrol had a deflected shot deflected wide at the other end before Sluga was forced to push Ayew’s header from the corner around the post with 59 minutes gone.
From the resultant corner, Borja’s header was cleared to the edge of the area where Byers’ goal-bound effort was again deflected wide. The ball found Byers again from the corner and he fired inches over on the hour.
It sparked a great spell for the Swans as Celina found Carroll with a neat flick, but his cross was cleared inside the six-yard area.
Luton made a double substitution in the 61st minute in a bid to get back in the game as Jacob Butterfield and Luke Berry were replaced by George Moncur and Ryan Tunnicliffe.
As the Swans pressed again, they were nearly caught on the counter as Cornick raced clear in the 69th minute. Woodman came racing out of his goal but could not prevent Collins getting passed him before pulling the ball back for substitute George Moncur. But his shot was blocked, and the Swans breathed a huge sigh of relief.
Shinnie headed another great chance over straight after before the visitors regained possession and their composure.
A double substitution followed I the 75th minute as Wayne Routledge and Jay Fulton replaced Borja and Carroll respectively.
Tempers flared in the 78th minute when Byers went down inside the area under a challenge from Matty Pearson. The referee booked Byers for simulation as Pearson reacted angrily and both sets of players got involved.
It was the large away support who were celebrating in the 82nd minute though as Andre Ayew opened the scoring right in front of them.
Celina played in Fulton to the by-line and his deflected cross fell perfectly for Ayew to chest in his 10th goal of the season from close range.
And that proved the difference as the Swans moved into the play-off spots.
Luton Town: Simon Sluga, Sonny Bradley (capt), Matty Pearson, Luke Berry (Ryan Tunnicliffe, 61), Andrew Shinnie, Harry Cornick (Callum McManaman, 74), Pelly-Ruddock Mpanzu, James Collins, Luke Bolton, James Bree, Jacob Butterfield (George Moncur, 61).
Subs (not used): Marek Stech, Kazenga LuaLua, Lloyd Jones, Alan Sheehan.
Swansea City: Freddie Woodman, Connor Roberts, Ben Wilmot, Mike van der Hoorn, Jake Bidwell, Matt Grimes (capt), George Byers, Tom Carroll (Jay Fulton, 75), Andre Ayew, Bersant Celina (Kyle Naughton, 87), Borja Baston (Wayne Routledge, 75).
Subs (not used): Kristoffer Nordfeldt, Brandon Cooper, Jay, Barrie McKay, Aldo Kalulu.
Referee: David Webb
Attendance: 10,062