Romelu Lukaku netted twice in the first half to put the visitors in an assured position, before Paul Pogba wasted a chance to seal the match when his penalty was saved by Joe Hart.
Substitute Marcus Rashford was given a straight red card for a clash with Phil Bardsley, but Burnley couldn't make their man advantage count.
Check out our breakdown of the game below.
Key Talking Point
It might only be September 2nd but this was Burnley's ninth game of the season already and that seemed to show, particularly in the first half. Sean Dyche's side, usually high on intensity, looked tired and sluggish for much of the match.
The Europa League has sometimes been called a poisoned chalice for smaller teams, who lack the depth to manage the demands of both the European competition and the Premier League, and it looked that way today for Burnley.
It remains to be seen, having been knocked out of the tournament now by Olympiakos, if they can recuperate and bounce back in the next few weeks.
Starting XI: Hart (7), Bardsley (4), Tarkowski (6), Mee (5), Taylor (5), Lennon (6), Westwood (5), Cork (5), McNeil (7), Hendrick (4), Wood (7).
Substitutes: Vokes (6), Barnes (N/A), Vydra (N/A).
STAR MAN - It's often a thankless task to be the lone striker when facing a much tougher opposition, yet Burnley striker Chris Wood continued to crop up and challenge for anything that came his way.
Still looking for that first goal of the season, the New Zealand international was held and hassled throughout but continued to be an outlet for Burnley as manager Sean Dyche will have asked.
WORST PLAYER - Jeff Hendrick didn't appear to be on the pitch for majority of this game until he was substituted, with the Republic of Ireland international failing to make any kind of real impact on the match.
Key Talking Point
For all the pressure being thrown at Jose Mourinho and his defensive players in the week, his attacking players certainly had no worries.
Alexis Sanchez, Jesse Lingard and Romelu Lukaku were a seamless attacking threat for the entirety of the first half (with the Belgian netting twice), even if they were facing an uncharacteristically slow and lethargic Burnley side.
Paul Pogba ran the show - though shouldn't be let off the hook for an awful penalty - and together the attack carried Manchester United to a crucial win and set them back on the right path with their form.
Starting XI: De Gea (7), Valencia (6), Smalling (7), Lindelof (6), Shaw (7), Fellaini (6), Matic (6), Pogba (8), Lingard (8), Lukaku (9), Sanchez (7).
Substitutes: Rashford (2), Herrera (N/A), Bailly (N/A).
STAR MAN - Romelu Lukaku was unplayable at times up front, bullying the usually strong Burnley defence. His opponents seemed leggy and lacking their usual intensity, and the Belgian took full advantage.
His brace was well-taken, even if the second goal was perhaps a little fortunate. The only criticism of him today is that he should have probably had more. There are definitely worse negatives to have.
Good: Romelu Lukaku scored twice today— James Tyler FC (@JamesTylerESPN) September 2, 2018
Bad: Romelu Lukaku should probably have scored like six today
WORST PLAYER - There was a real lack of maturity shown by Marcus Rashford in his sending off that perhaps reminded everyone that it is still a young man out there despite his successes. Phil Bardsley had a kick at him, yes, but he's wise and experienced at winding up attackers that way, and Rashford just wasn't controlled enough to keep his emotions in check.
Burnley's next Premier League fixture is a trip to face newly-promoted Wolves on Saturday, who have started the season well and look a formidable opposition to face.