The Clarets, who were unbeaten in their previous four visits to the Theatre of Dreams, were on course to stretch that sequence after James Tarkowski had cancelled out Mason Greenwood' s second half opener.
However, the United teenager was aided by a wicked deflection off Jack Cork's heel as he fired the home side back into the lead in the 84th minute.
Sean Dyche's side were then caught on the break as substitute Edison Cavani tapped the ball home from close range in stoppage time.
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Sean Dyche's side were then caught on the break as substitute Edison Cavani tapped the ball home from close range in stoppage time.