The home side scored twice in the opening quarter of an hour through Welsh international Sam Vokes and Iceland's Johann Berg Gudmundsson, though goalkeeper Steve Mandanda was at fault for the second.
Substitute Connor Wickham pulled a goal back for the London side with his first touch and Christian Benteke converted a penalty for handball.
Burnley, however, secured a comfortable mid-table spot with a dramatic winner in stoppage time by Ashley Barnes.
(Reporting by Steve Tongue, editing by Ed Osmond)
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