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.
1. Bailey Peacock-Farrell 7.4
Blameless for United's three goals and made a string of saves. Flicked a Pogba header around the post, turned a Rashford strike over the bar before saving second half efforts from Fernandes and McTominay. Photo: Gareth Copley
2. Matt Lowton 6.7
Shoulders much of the blame for United's opener, needlessly diving in on Rashford. Failed to close the gap between himself and the player in possession, in threatening positions, but distribution was a positive. Photo: Gareth Copley
3. Charlie Taylor 6.9
Stuck to his task admirably when Wan-Bissaka, Greenwood and Fernandes combined down United's right hand side. Didn't spot Greenwood lurking in behind for the opener and could maybe have got closer to the teenager for the second. Photo: Jon Super
4. James Tarkowski 8
Caught out with the pace of United's forward line once or twice, but otherwise defensively sound. Movement for his goal was impressive and the header was even better. Photo: Jon Super