Scott Parker’s side showed plenty of character to stay in the game during the second-half after suffering a nightmare end to the first 45 minutes.
Despite playing well for the first half an hour, Burnley suffered a soft first goal from a direct corner before being reduced to 10 men through Lucas Pires' straight red.
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An Anthony Gordon penalty in first-half stoppage-time looked to set up a miserable second period, but the Clarets reduced the deficit through Zian Flemming before glancing a header agonisingly wide of the far post with the last action of the game.
Here’s how Burnley’s players rated:
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13. SUB: Josh Laurent - 6/10
For Florentino, 64 Came so close to grabbing an unlikely last-gasp equaliser at the death. Put in a good shift in a slightly more advanced role to aid Burnley’s off-the-ball work. (Photo by George Wood/Getty Images) Photo: George Wood
14. SUB: Jaidon Anthony - 6/10
For Bruun Larsen, 85 Tried to make things happen during the final stages as Burnley battled desperately for a late point. (Photo by George Wood/Getty Images) Photo: George Wood
15. SUB: Zian Flemming - 7/10
For Broja, 85 Reduced the deficit from the penalty spot with his fifth goal in his last seven games. Perhaps glanced the ball off Josh Laurent's head for that last-gasp chance. (Photo by George Wood/Getty Images) Photo: George Wood
16. SUB: Marcus Edwards - 6/10
For Ekdal, 90 Provided the tantalising cross that almost brought about that unlikely last-gasp equaliser (Photo by Molly Darlington/Getty Images) Photo: Molly Darlington