A draw with Preston North End last weekend, another promotion challenger, boosted the Hatters' confidence in South Wales, and they should have led in the first half when Harry Cornick fired over.
However, James Collins struck with 20 minutes to go, squeezing his header into the corner beyond the unfortunate Freddie Woodman, before substitute Jordon Garrick was dismissed for attempting to hit the forward.
The result leaves Luton two points from safety but moves Swansea down the table into tenth, and four points off the play-offs.
The Swans were unchanged from last weekend's 3-0 win at Middlesbrough, though Joe Rodon returned to the bench after resuming full training this week. An ankle injury to Kyle Naughton sidelined him last time out, but he was a substitute once again for Steve Cooper's side.
Glen Rea brought up his 150th appearance for the Hatters as they made four changes from the 1-1 draw against Preston. Captain Sonny Bradley returned alongside Pelly-Ruddock Mpanzu, Jacob Butterfield and Elliot Lee, who started for the first time in the league since August. Callum McManaman was benched despite his late equaliser seven days ago.
Luton started the brighter of the two, contrary to what the table would suggest, as Dan Potts forced a corner down the left channel, before Lee won a free-kick overhit by Butterfield, the only highlights of an uneventful opening ten minutes.
Potts continued the early pressure on the Swansea goal, this time Butterfield's delivery was better and met by the full-back who found the side netting.
The hosts nearly took the lead moments later as Andre Ayew fired towards Simon Sluga after Jake Bidwell's superb pass, but the Croatian was on hand to turn the effort behind.
This was an end-to-end affair and appeared to live up to more than December's reverse fixture, a narrow 1-0 win for Swansea at Kenilworth Road, as Cornick broke from the corner, cut inside and fired over, a huge opportunity.
The Swans started to pick up the pace as Jay Fulton, who appeared to fall on his head minutes before, saw his effort blocked by Cameron Carter-Vickers. Rhian Brewster's deflected header then dipped over the crossbar just before the break.
Rea and Butterfield were booked for fouls on Aldo Kalulu and Conor Gallagher respectively, though the visitors kept surprising Swansea with their attacking play as Potts threatened down the left flank again. Watford loanee Ben Wilmot then fouled the latter of the cautioned men, receiving a yellow card of his own.
The hosts started the second half well, their best chance of the opening exchanges came courtesy of Ayew, whose shot deflected against the feet of Bradley before Sluga gratefully gathered.
Mpanzu's shot from 25-yards brought the visitors back into the contest on the hour mark, but Woodman was unmoved to stop this. The to-and-fro nature of the contest didn't stop, Matt Grimes' searching delivery found Ayew, who nodded harmlessly past the post.
Luton didn't take their chances in the first half and were nearly made to pay the price, being pinned back for most of the second but thankful for the reactions of Sluga who added Ben Cabango to the list of frustrated Swans to be kept out after his close-range header.
However, against the run of play, but to their credit for their courageous offensiveness, the struggling Hatters took a shock lead in the 72nd minute. Izzy Brown's pinpoint cross found the head of Collins and his effort saw off the best efforts of Woodman.
Collins turned attack into defence and turned away a dangerous cross at the other end as Swansea had to throw the kitchen sink at their visitors, in order to avoid being cut adrift from the playoff spots.
The goalscorer was joining Sluga as the most frustrating men in Wales this afternoon, as both combined to deny the hosts once more. First the latter reacted well again, this time to a Grimes free-kick, before Collins headed clear yet another corner.
Clearly that frustration got to Garrick, who came on at the break, as he aimed a punch at the Hatters goalscorer and was subsequentltly shown his marching orders by Andy Woolmer. Mpanzu entered the book for his involvement in the incident but this didn't impact much, if anything at all. Despite the results of both sides helping each other last weekend, clearly all respect had been left in the changing rooms.
Nine minutes of added time followed the goal, melee and increased number of substitutions available to sides post-lockdown, but despite any initial fears, the Swans couldn't make this count and Luton recorded an impressive victory on the road to boost their hopes of survival.
The managers' verdict
Cooper was immensely disappointed with his side's defeat, telling the Swansea website: "We have only got ourselves to blame, that's for sure.
"I did not like our attitude towards the game once it started. We waited for the person next to us to get the job done rather than doing it ourselves."
The stats that matter
As mentioned, Luton hadn't won at Swansea for 18 years, but have now ended that run as well as the home side's four consecutive victories over the Hatters.
As for the game itself, this wasn't a smash and grab by any means. The visitors were the better team in the first half but in the end were subject to an onslaught after the break, with the Swans enjoying 63% possession but they failed to make it count.
Ultimately, the only stat that matters is goals: 1-0 to Luton.
What does it mean?
For a side challenging for promotion, a home fixture with the side bottom of the league is a mouth watering prospect. However this didn't quite work in Swansea's favour, as they fall to tenth and their one-point gap from from the top six expanded to four, owing to Cardiff City's impressive 3-1 away win over Preston, whom they leapfrog into the playoffs.
For Luton, their chances of survival have been granted a huge boost. They move within just two points of Middlesbrough, who appointed Neil Warnock as manager this week but occupy the final safety spot. They also move off the foot of the table after Barnsley drew with Millwall.
Swansea travel to South London on Tuesday for a huge clash with Millwall, who are still very much in contention for a playoff spot despite that goalless draw.
Man of the Match - James Collins
The forward was a constant thorn in the side of the hosts this afternoon, not just scoring the winning goal but swapping attacking duties for those in defence, when it mattered most, and coming up successful in both areas.
Swansea City: Woodman; Roberts, Wilmot, Cabango (Routledge 86), Bidwell; Fulton (Byers 58), Grimes; Kalulu (Garrick 46), Gallagher, Ayew; Brewster.
Subs not used: Mulder, Naughton, Rodon, Guehi, Dhanda, Dyer.
Luton Town: Sluga, Potts (Cranie 90), Bradley, Pearson, Lee (Bree 69), Cornick (Hylton 90), Rea, Mpanzu, Carter-Vickers, Collins, Butterfield (Brown 59).
Subs not used: Tunnicliffe, McManaman, Shinnie, Moncur, Shea.
Referee: Andy Woolmer.