Swansea 3 Burnley 2Andre Gray struck twice but couldn't prevent the Clarets tasting further defeat on the road in a dramatic finish in South Wales.
Gray converted a controversially awarded penalty to cancel out Fernando Llorente's early header.
And the fast-maturing striker took his tally for the season to nine with an excellent second-half finish which set up the possibility of a first away win of the season and first victory for Burnley at Swansea since 1995.
But the Clarets' rearguard action was punctured by Martin Olsson and then again deep in stoppage time by Llorente, who grabbed his own second goal of a gripping game to leave the Clarets empty-handed in agonising fashion.
The Clarets welcomed back Jeff Hendrick from suspension but lost the services of goalkeeper Tom Heaton on the eve of the game after he was struck down by illness overnight, meaning a third Premier League start for Paul Robinson.
The former England international had been restricted to kicking duty in the opening 10 minutes but in a two-minute blitz Swansea twice hit the woodwork and then went in front.
Defender Alfie Mawson was the unlucky individual at first when he headed a free-kick from Gylfi Sigurdsson againt the bar and then struck a shot against the foot of a post as the Clarets were unable to clear.
But Swansea's pressure did pay off when Leroy Fer - the Swans' match-winner at Turr Moor in August - got away down the right-hand side and delivered a magnificent cross which Llorente powered into the back of the net with a thumping header.
Swansea had won all three previous Premier League meetings between the two clubs 1-0, but this one didn't stay that way for long.
Gray sparked the fightback with a smart turn and deflected shot which was touched around the post at full stretch by Lukasz Fabianksi.
Swansea conceded a second corner off the back of that one and when Robbie Brady delivered, the ball hit the raised arm of Sam Vokes as the Wales international stretched to try and connect with a header.
Referee Anthony Taylor mistakenly pointed to the spot and Gray kept his cool following lengthy Swansea protests to drive home the eighth goal of an increasingly productive campaign.
It was a fortunate break and Burnley had to work hard to make the most of it as Swansea responded positively in pursuit of a third successive home win.
Dutch international Lucianio Narsingh was a frequent threat down the right-side and when Olsson burst clear down the opposite flank for the Welshmen, it took a strong hand from Robinson to deny his former Blackburn Rovers team-mate.
Sigurdsson then fired just wide before Llorente sent a looping header against the Burnley bar from an Olsson cross as the home side strove to regain the lead.
The influential Sigurdsson also saw a free-kick beaten out by Robinson with the final touch of an eventful first half and the break did little to stem the Swans' momentum, Sigurdsson again going close as he beat Robinson but not Ben Mee as the covering Clarets' captain cleared off the line two minutes after the re-start.
Llorente was next to threaten as the big striker glanced another Olsson cross just wide of the far post.
But Burnley battled to stay in the contest and with just over an hour gone their doggedness paid off as Gray struck again to put the Clarets in front.
Vokes was again involved, this time with a knock-down from Matt Lowton's ball forward to find his strike partner just inside the penalty area.
Gray still had plenty to do and the Clarets' leading scorer - an excellent outlet for the visitors all afternoon - did it superbly, allowing the ball to run across his body as he swivelled and planted a low shot across Fabianski and into the bottom corner.
It was a lead, though, that was to last just eight minutes as Vokes just failed to get a toe-end to Gray's low cross to potentially kill the game off and the Swans responded to make it 2-2 with a fine goal of their own.
Sigurdsson was the architect with a clever flicked pass into the path of left-back Olsson, who burst into the box to hit a fierce shot into the top corner.
Olsson then thumped a volley just over the bar as a pulsating game entered its last 10 minutes with the Clarets scrapping to hold onto a point, Ashley Westwood and James Tarkowski coming on for Vokes and a limping Joey Barton.
It was all hands to the pump to prevent Swansea from continuing their climb away from relegation danger as Dyche's men - despite being second-best - dug deep to maintain their own progress.
The ever-resilient Mee blocked another goalbound effort from Tom Carroll but in injury time the Clarets' defences were breached for the decisive time as Swansea grabbed a vital victory right at the death to aid what looks - on this evidence - like a winnable survival fight.
Llorente put a hand into the back of Mee as he rose to meet Carroll's swinging cross but this time the verdict went the Swans' way as the big Spaniard headed back across Robinson, whose out-stretched finger tips were just not enough to keep the ball out and preserve a priceless point.
Burnley: Robinson, Lowton, Keane, Mee, Ward, Boyd, Barton (Westwood 80), Hendrick, Brady (Arfield 66), Vokes (Tarkowski 80), Gray. Substitutes: Pope, Flanagan, Darikwa, Agyei.
Booked: Ward, Mee
Swansea: Fabianksi, Naughton (Rangel 89), Fernandez, Mawson, Olsson, Narsingh (Ayew 73), Cork, Carroll (Amat 90), Sigurdsson, Fer, Llorente. Substitutes: Nordfeldt, Britton, Routledge, Borja.
Booked: Fer, Cork
Referee: Anthony Taylor