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.
Here are the RATINGS.
9. Ashley Westwood 7.5
Pressed the ball hard and never stopped running, which allowed the Clarets to slow the play down. That proved hugely beneficial for the visitors. Tidied up the bits and pieces around the box and looked to get the ball moving in midfield. Photo: Stu Forster
10. Josh Brownhill 8.1
Worked admirably well in transition, getting up to support Wood in a more advanced role, while working hard to get back behind the ball when United broke up play. Covered an incredible amount of ground and built up a good understanding with Burnley's lone striker. Photo: Gareth Copley
11. Chris Wood 8.2
Employed as a lone striker, but was hugely effective for the Clarets. Had a goal chalked out after 14 seconds, would constantly get the better of Maguire and Lindelof in the air, held up the ball well and picked up some good goal-scoring positions. Photo: Gareth Copley