Manchester, which has not lost in the Premier League for six meetings (4 wins, 2 draws), goes from seventh to sixth place, tied on points with West Ham (5th) who won 4-1 at Watford Tuesday.
But the Red Devils are still nineteen lengths behind (and two games less) on the leader Manchester City.
Burnley for his part remains eighteenth after a day cut off by two meetings because of Covid: Everton-Newcastle and Leeds-Aston Villa.
Under the eyes of his legendary ex-coach Sir Alex Ferguson, who will celebrate his 80th birthday on Friday, Manchester United opened the scoring in the 8th minute, by Scottish midfielder Scott McTominay, with a flat foot at the entrance from the surface.
Two minutes earlier, Cristiano Ronaldo had had a first big chance, on a deep ball from Luke Shaw, but his shot from the flat foot in the middle of the race went over the goal of Wayne Hennessey.
Dominating in the rain of Old Trafford, the Mancuniens doubled the score in the 27th minute thanks to a goal against his camp from Burnley defender Ben Mee, who deflected a shot from Jadon Sancho in his goal.
Then the inevitable Ronaldo went there with his personal touch: his eighth goal of the season in the Premier League after receiving a ball pushed back to the post by Hennessey after a nice strike from McTominay (35th).
While at that moment the worst was to be feared for Burnley, the 34-year-old striker Aaron Lennon reduced the score on a counter-attack punctuated by a cross shot at the ground (38th).
Four goals in the first period … then nothing in the second!
Manchester United, who played without their injured French Paul Pogba and Anthony Martial (Raphaël Varane came into play in the 66th minute), will receive Wolverhamton on Monday January 3 for their first match of the year.