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...
5. Ben Mee 8
A characteristically blood and thunder display from the Burnley captain. Battered, bruised, but never beaten. Puts his head in where it hurts, always getting on the end of crosses in his own penalty, and cut out a number of dangerous passes.
6. Johann Berg Gudmundsson 6
Bright in the first half, sitting balls up towards United's left hand side for Barnes to pin Wan-Bissaka, and got back behind the ball to limit United's creativity down that side of the pitch. Tired in the second half and struggled to get into the game.
7. Robbie Brady 6
Fluffed a glorious chance to get Burnley off to the best possible start, but showing signs of getting back to his 'old' self. Neat and tidy in possession, took the sensible option when on the ball, and worked hard up and down that flank.
8. Josh Brownhill 6
Stuck to his defensive duties well, covering the full back areas whenever needed, and kept United's dangermen at arm's length. Fatigue crept in during the second half, which enabled Fernandes and Pobga to pick up dangerous pockets and get in behind.