The Clarets went out of the Premier League Cup as a first-half winner from Kenji Gorre denied Burnley the point they needed to reach the knock-out stages.
Burnley knew avoiding defeat would see them clinch second spot in Group D and a place in the 16.
But despite an impressive display against the holders, a solitary strike from Gorre – the son of ex-Barnsley and Huddersfield midfielder Dean – was enough for Swansea to avenge a defeat in South Wales in the reverse meeting and take them through by the skin of their teeth.
Michael Duff’s side gave everything to try and force the equaliser that would have seen them advance along with group winners Newcastle United.
And after Jordan Barnett had missed an excellent chance, Ntumba Massanka crashed a shot against the underside of the bar with 10 minutes to go as the Clarets came up just short.
The Clarets fielded their youngest line-up so far in the competition with Barnett making his first start and Jordan Cropper getting his first taste of under-23 action.
And Cropper impressed at right-back – alongside captain Arlen Birch who led by example – as well as delivering a dangerous long throw which looked the Clarets’ most likely route to goal in the first half.
Christian N’Guessan headed wide after Massanka had flicked on one of Cropper’s deliveries from the best of the openings, while Swansea were given only limited sights of goal at the other end.
Conor Mitchell made a smart early save from Jordan Garrick, pushing his effort onto the post, but Burnley’s one defensive lapse proved crucial with 36 minutes gone.
Swansea were allowed to ferry the ball across the pitch too easily and Gorre showed an assured touch to drill the ball across Mitchell into the far corner to claim the goal that maintained the Swans’ grip on the trophy.
In freezing conditions and flurries of snow on a pitch that needed to pass a pre-match check, things heated up even further after the break with both sides knowing one goal would change everything.
Mitchell made another important block to deny Liam Cullen the second goal that would have made things more comfortable for the visitors.
But most of the attacking was done by the Clarets as they pressed to keep their hopes alive.
Burnley have found goals hard to come by - not scoring since their 2-0 win at Swansea in January.
But they carried a threat here, the lively Jackson crossing for Barnett to lift an effort over the top with just under 20 minutes to go when the youth-team winner looked set to score his first senior goal.
And then with time running out, only the width of the crossbar denied Duff’s men a place in the next round and a reward for their efforts in winning at Newcastle and Swansea earlier in the competition when Massanka’s shot on the turn thundered against the underside of the bar and away from goal.
Burnley: Mitchell, Cropper, Birch, Wilson, Glennon, Jackson, Nabi (Koiki 90), N’Guessan, Barnett (Chakwana 72), Wood, Massanka. Substitutes: Stone, Younger, Thomas.
Swansea City: Bena, Davies, Harries, Cabano, Lewis, Garrick, Evans, Dulca, Gorre, Biabi, Cullen. Substitutes: Darame, Zabret, De Boer, Copper, Baker-Richardson.
Referee: Peter Wright