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.
5. Ben Mee 7.9
Lost count of the amount of times he won headers on his penalty spot, blocked efforts and stuck a leg out to cut out crosses. Strong performance from the skipper, which helped limit United's success in the final third. Photo: Stu Forster
6. Johann Berg Gudmundsson 6.1
Almost helped the Clarets get off to the best possible start when delivering the ball for Wood inside the opening minute. However, contribution was limited thereafter. Gets into good positions, but requirement to cut back onto his left his downfall. Photo: Gareth Copley
7. Dwight McNeil 7.2
Worked hard to get behind the ball in support of Taylor as Wan-Bissaka and Greenwood threatened. Tidy in possession, saw a goalbound effort blocked by Brownhill and won the corner that led to the equaliser. Photo: Gareth Copley
8. Jack Cork 7.7
Broke up the play well on the edge of the penalty area, stifling United's moment on a number of occasions. Carried Burnley out of trouble when needed, composed on the ball, but unfortunate to see Greenwood's effort take a nick off his heel.. Photo: Gareth Copley