The Clarets fell behind early on courtesy of a Fernando Llorente header, with the home side dominating throughout the first half. However, a controversial penalty in the second half gifted Sean Dyche's side a way back into the match, with Anthony Taylor pointing to the spot following a corner.
The referee deemed there to be a handball from a Swansea defender, however replays showed it was in fact Sam Vokes whose arm made contact with the ball. Nonetheless, Andre Gray made use of the opportunity and brought the scores level.
When Gray produced his second of the game moments later, turning well and volleying into the bottom corner, it seemed as though Burnley could record their first away win of the Premier League season.
However, justice was done in the end, when a sumptuous goal from Martin Olsson was followed up by another header from Llorente to give the home side an important three points in their quest to avoid the drop.
Goalkeeper and defence
Paul Robinson - 6 - Nothing he could do about any of Swansea's goals. Llorente delivered his headers with pinpoint accuracy while Martin Olsson's powerful close range strike was never going to be stopped.
Matt Lowton - 6 - Produced a terrific ball to Sam Vokes in the build up to Andre Grey's second goal, but struggled on a few occassions defensively.
Will Keane - 6 - Produced a few timely interceptions to keep his side in the game and dealt well with Swansea's crosses into the box.
Ben Mee - 4 - Bullied and barged by Lllorente throughout the match, and had no chance of nullifying the Spaniard. Arguably fouled for Swansea's winning goal but he had to be stronger today.
Stephen Ward - 5 - Had a torrid afternoon against Narsingh, whose pace proved to be a constant menace to the Burnley defence.
George Boyd - 6 - A very quiet game from Boyd, who's usual energy and verve was hard to spot due to Swansea's dominance in the middle of the park. Helped his side out defensively to his credit.
Joey Barton - 6 - Another quiet performance from Barton, although he did combine well in midfield on a few occasions. Not one of his best displays.
Jeff Hendrick - 6 - Couldn't deal with the physicality and energy of Jack Cork in midfield. Not one of his best displays by any means.
Robbie Brady - 5 - Came off during the second half after making little impact on the match. Wasn't afforded any time or space to make his mark on the match or produce any moments of quality for Vokes to make the most of.
Sam Vokes - 5 - Didn't manage to have much impact on the match, bar giving his own side a penalty thanks to some controversy.
Andre Gray - 7 - Burnley's biggest goal threat throughout the match, showing a great turn of feet before wrapping his volley into the bottom corner to put his side ahead.
Scott Arfield - 5 - Little impact despite replacing Brady just after the hour mark.
Ashley Westwood - 6 - Barely involved after coming on late on, completing just one pass. Couldn't help Swansea preserve a point.
James Tarkowski - 6 - Same as above.
Ultimately, Burnley were never in control of the game, even after going ahead through Gray's second.
The result will raise some eyebrows as to whether their place in the Premier League for next season is entirely secure after another defeat on the road, with Liverpool next up for Sean Dyche's side at Anfield.