Joao Felix broke the deadlock with an arrowed drive into the far corner six minutes into the second half.
Browsing: James Tarkowski
It looked set to be a frustrating outing for the Red Devils as Greenwood’s 48th minute opener was almost instantly cancelled out by James Tarkowski for Burnley, who had not lost on their previous four visits to Old Trafford.
Ralph Hasenhuttl’s side moved 10 points clear of the relegation zone with eight games left after a dramatic fightback at St Mary’s.
James Tarkowski had given the home side a first half lead.
The Clarets are keen to keep the England international at Turf Moor, although talk of a £40 million move elsewhere means that an exit remains a possibility.
Arsenal midfielder Jack Wilshere is recalled, while Lewis Cook could make his debut after previously being included in the squad in November.