Paul Pogba scored the only goal of the game in the 71st minute when steering a Marcus Rashford cross through the legs of Nick Pope.
It was Burnley's fifth defeat on the bounce at home to the Red Devils and the 12th time in their last 13 meetings at Turf Moor where they'd failed to score.
Here are our RATINGS as Burnley's first league game of 2021 ends in defeat...
1. Nick Pope 6
Made a number of routine saves to keep out Fernandes and did well to foil Martial with his legs when the Frenchman had a sight of goal late on. Beaten by Maguire's header, which was chalked off, and was unfortunate with Pogba's winner.
2. Matt Lowton 6
Almost gifted Cavani an opening goal with a poor clearance from a corner and unfortunate that his touch diverted Pogba's volley past Pope. Didn't really put a foot wrong, defended narrowly and resolutely, forcing United wide, and played a number of effective diagonals.
3. Erik Pieters 6
Versatile, reliable. Gave very little away in the first half when standing up to Rashford and keeping the United forward on the outside. Tidy with the ball at his feet, but struggled when Pogba started to pull the strings and feed the ball into the inside channels.
4. James Tarkowski 7
Defended exceptionally well, both in and around the area, with his challenge to deny Martial his stand out moment. Restricted United's front four to speculative efforts, but squandered a huge opportunity to steal a point late on.