Paul Pogba gave United the lead and put in a dominant performance to send his side to the top of the league table. Pogba's goal started with him winning the loose header, Bruno Fernandes pinging the ball out wide for Marcus Rashford, who found Pogba and the French midfielder volleyed it past Nick Pope.
Sean Dyche's side defended resiliently for the majority of the game and were only outdone by a moment of true brilliance from Pogba.
Story of the game
Right from the off it was apparent that the Clarets were happy for United to have the ball so long as they were far enough from goal. Dyche's men also had the game plan of attacking Aaron Wan-Bissaka with long balls directed over the right-back's head. This was not the first time that tactic has been deployed against Wan-Bissaka.
Cavani was then running in on goal when Robbie Brady brought him down just before he entered the penalty area. Referee, Kevin Friend, handed Brady a yellow card initially but VAR took a look at the Luke Shaw challenge prior to the Brady tackle and gave Shaw a yellow card instead of Brady.
In a brief moment of madness, Shaw escaped a red card as well as Brady who brought down Cavani in a goal-scoring opportunity.
Harry Maguire then had a goal ruled out by the referee after an apparent foul in the box. It was harsh from Friend and Maguire made his frustrations heard.
At half-time the teams went down the tunnel level.
A wonderful move by United early in the second half nearly gave the them the lead. Martial's bouncing cross found Cavani who could not find the right connection to send it goalwards.
United found their rhythm around the hour mark and were knocking on Burnley's door. But Dyche's defence remained resilient and focused at the back.
Cavani had another attempt with a curling left-footed strike inside the box but it was well wide. Despite the striker using his weaker foot, he should have done better.
It was Pogba who took the game to Burnley and buried home an excellent volley. It was Rashford's cross which was perfectly placed for Pogba to smash it first time past Burnley 'keeper, Pope.
Matej Vydra almost equalised incredibly late on but fired his effort way over the bar.
United top in January for first time since 2013
Solskjaer and his side will no doubt understand where they are in the table but he will want his side to just want to deal with one game at a time and not get ahead of themselves. They may be top of the table for the first time since 2018 and top for the first time at this stage of the season since 2013 - but as Solskjaer has said previously, it does not matter where they place in January, it matters where they finish in May.