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Chris Wood scored a dramatic stoppage-time penalty to secure a draw with Wolves as both sides failed to further their European ambitions.
Wood fired his spot kick into the top corner in the 96th minute following a Matt Doherty handball.
Raul Jimenez had put Wolves ahead with a superb volley from the edge of the box after 76 minutes.
It looked like Nuno Espirito Santo's side had closed the gap on a Champions League place to one point, albeit having played a game more than rivals Leicester City and Manchester United.
However, Burnley, who have Europa League hopes of their own, almost levelled when Wood nodded wide of an open goal with keeper Rui Patricio stranded.
And moments later a hopeful cross struck Doherty on the arm and the New Zealand international made no mistake with his penalty.
It means Burnley sit ninth in the Premier League, whiles Wolves remain sixth and three points off the top four.
Action and reaction from Wednesday's Premier League gamesClarets show resolve to snatch point
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Burnley boss Sean Dyche says he was once told by Ian Holloway his side have a "strong jaw", and they had to show all their resilience to earn a point against Wolves.
Jimenez looked to have landed a knockout blow when he struck with 14 minutes to play but Burnley have lost just one of their past 14 league games for a reason.
The hosts had offered little in way of an attacking threat before added time, half chances for Jay Rodriguez and Josh Brownhill not troubling goalkeeper Patricio.
But then came two glorious opportunities for Wood, who showed composure to convert his spot-kick after a long VAR check having earlier turned his header wide.
"I went all game with one chance I didn't put away so it's nice to get another chance and get a point," said Wood afterwards.
"It was great mentality from Woody," Dyche added. "When he missed that chance, it shows you what a strong character he is to step up and take the penalty and finish it in the style he did."
It means the Clarets are unbeaten in six games and still hold an outside chance of making the Europa League.
Dyche, however, will no doubt be satisfied if they record a top-half finish this season.
Wolves let it slip late on
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Wolves' hopes of qualifying for the Champions League this season were dealt a significant blow when Manchester City's two-year European ban was overturned on Monday, taking fifth place out of the equation.
They only had themselves to blame for dropping points on Wednesday, however, after failing to convert their attacking dominance into more goals.
This was the 26th time Wolves have failed to score in the first half of a league game this season, with Jimenez heading straight at Pope and Romain Saiss flicking Joao Moutinho's dangerous corner over the crossbar.
Adama Traore, a constant threat on Wolves' right, flashed an effort wide of the far post shortly after the break and it took a wonderful piece of skill from Jimenez to put them ahead.
After another threatening Traore run, Doherty's effort deflected into the path of Jimenez and the forward opportunistically struck it first time on the volley for his 17th Premier League strike of the season.
It was Wolves' 36th goal in the second half this season - three times more than they have netted in the first 45 minutes of games.
It is also almost a year since Wolves began their season, a 2-0 second qualifying-round win over Crusaders in the Europa League on 25 July 2019, and this was their 55th game in all competitions.
Perhaps that lengthy campaign took its toll late on at Turf Moor, when the visitors looked set for another impressive win before Burnley's last-gasp equaliser.
Man of the match - Adama Traore (Wolves)
Prolific Jimenez notches again - the stats
Since beating Burnley in their first Premier League meeting under Nuno Espírito Santo (1-0 in September 2018), Wolves have gone winless in the three meetings since (D2 L1).73% of Wolves' goals in the Premier League this season have been scored in the second half (36/49); the highest percentage of any side.Raul Jimenez is the first Wolves player to reach 17 league goals in the top flight in a single season since John Richards in 1975-76 (also 17).Only Mohamed Salah has scored more Premier League goals in 2020 than Wolves' Jimenez (9).Chris Wood's equaliser from the penalty spot was Burnley's only shot on target in the game, coming in the 96th minute.'They never seem to know when they're beaten'
Burnley manager Sean Dyche: "It's another interesting twist that goes our way, and we worked to make it do so. It was another good finish to a strange game.
"The Premier League is very unforgiving, and to put that run of games together and put points on the table in that run as well, coming out of a tough run when it started, I can't speak highly enough of this group.
"They continue to give more and more to the cause. Players come in and they deliver. There is a relentlessness to what they're doing. They never seem to know when they're beaten."
Wolves boss Nuno Espirito Santo: "Every goal you concede is hard to take and if it is in the final moments it is even harder.
"I think we played good, we were well organised on the pitch and it was a good game. The team were always in control, we managed the game well. You try to prepare good but after that sometimes justice in the game doesn't exist.
"But this is not the point of football. The point of football is to compete, play well, and the boys did this today."
What next?
Burnley visit relegated Norwich on Saturday (17:30 BST), while Wolves host Crystal Palace on Monday (20:15).