The visitors have been a thorn in United’s side at Old Trafford since 2015 and beat the Reds last season here.
Sean Dyche's side made it difficult again and after Greenwood scored early in the second half, they equalised through James Tarkowski.
But Greenwood was back on the scoresheet again with six minutes to go and half-time substitute Cavani wrapped it up in the added-on minutes.
FIRST HALF - EARLY SCARE, THEN STALEMATE
Burnley’s recent Premier League record at Old Trafford dominated pre-match talk with the visitors having won one and drawn three in their last four visits.