Chris Wood’s injury-time penalty kept the Clarets’ unbeaten home run going as Sean Dyche’s men refused to lie down.
A first Turf Moor defeat since New Year’s Day looked likely when Raul Jimenez scored with a superb 76th-minute volley to take Wolves closer to a European return.
But Burnley dug deep to keep alive their slim hopes of making it to Europe themselves with Wood keeping his cool to level from the spot following Matt Doherty's handbll as Sean Dyche’s battlers added another point to their collection and made it just one loss from 14 league games.
Six-and-a-half months after their last home defeat it looked as though the Clarets had finally run out of steam once Jimenez had scored his 27th goal of the season.
That cost Nick Pope a 15th clean sheet of the campaign on a night neither goalkeeper was over-extended.
But despite their struggle to make chances, the Clarets still found the resolved to get some reward and gain amends for the last-ditch penalty they conceded in a 1-1 draw at Molineux in August.
And Wood deserved great credit for blotting from his mind a missed opportunity moments earlier as the top scorer claimed his first goal since lockdown to get Burnley within three points of their record Premier League tally with two games to go.
Unsurprisingly, the Clarets were unchanged from the side that started out at Liverpool where they became the first to take a point from Anfield this season.
But the line-up didn’t survive as far as the break with Charlie Taylor picking up a knock to add to the injury list with half an hour gone – Erik Pieters swapping to a more familiar role at left-back and Johann Berg Gudmundsson continuing his comeback as the replacement on the right side of midfield.
The Clarets had just enjoyed their best spell of the half with Wood unlucky to see a goalbound strike hit team-mate Jay Rodriguez, who later volleyed over after collecting a trademark around-the-corner pass from Ashley Westwood and then almost profited from a superb ball on the break from Gudmundsson.
But Wolves showed why their goals-against record away from home is bettered only by the champions with a solid base which gave the Clarets little room for manouevre.
And despite playing their 55th game of the season – with at least one more Europa League tie still to come – the visitors were full of running when they went forward with pace.
They looked a constant threat, although Burnley’s back line again stood up to the test with Nick Pope not required to make a first-half save.
Wolves’ best chance was created by Daniel Podence – whose goal at Turf Moor two years ago helped Olympiakos knock the Clarets out of the Europa League – but from his cleverly floated pass, Diogo Jota put his shot on the turn into the side netting.
The Clarets were still in the game, though, and despite the departure of Rodriguez five minutes in, started the second half on the front foot.
Josh Brownhill went on the hunt of his first Burnley goal as he burst into the box following good work from Wood and it took a fine block from Ruben Neves to block the midfielder’s effort.
But the Clarets almost gifted Wolves an opener when a breakdown in communications between Pieters and Dwight McNeil - a growing presence at the other end of pitch - presented Podence with a shooting chance he dragged wide.
Burnley kept probing but with Kevin Long then requiring lengthy treatment after helping break up another Wolves counter-attack the Clarets were beginning to get stretched.
And with just over 15 minutes to go they cracked. The pacy Adama Traore led a break and even though Matt Doherty got lucky with a shot which was blocked by James Tarkowski, the quality of the finish from Jimenez, striking the loose ball first-time on the volley, was of the highest order.
Typically, the Clarets refused to lie down though and in stoppage time they rescued a point.
Wood had just missed a great heading chance from Matej Vydra’s knock-back but even deeper into stoppage time referee Mike Dean penalised Doherty for handling the ball with his arm raised as he challenged the striker and pointed to the spot.
And this time Wood didn’t err as he buried the opportunity with a perfectly placed effort into the top corner to claim his 12th goal of the season and keep the Clarets fighting on.
Burnley: Pope; Bardsley, Long, Tarkowski, Taylor (Gudmundsson 30); Pieters, Westwood, Brownhill (Brady 85), McNeil; Rodriguez (Vydra 51), Wood. Substitutes: Peacock-Farrell, Thompson, Dunne, Benson, Goodridge, Glennon.
Wolves: Patricio; Boly, Saiss, Coady; Traore, Neves, Moutinho, Vinagre (Jonny 84); Podence (Doherty 70), Jimenez (Dendoncker 84), Jota (Neto 78). Substitutes: Ruddy, Jordao, Gibbs-White, Kilman, Buur.
Referee: Mike Dean