United, who were the only side in the top flight without a goal in the opening 15 minutes of a game, went ahead in the eighth minute, against the run of play.
Matt Lowton had put the ball on a plate for Chris Wood in the minutes prior, after Dwight McNeil's attempt was blocked, but the New Zealander turned his header well wide.
The Red Devils capitalised as Cristiano Ronaldo's loose control from Mason Greenwood's pass rolled fortuitously to Scott McTominay who, using Ashley Westwood to shield the flight of the ball, finished past Wayne Hennessey with a first time shot.
Burnley, again, had the chance to find the net before United's second, when Johann Berg Gudmundsson's strike from Matt Lowton's delivery was blocked by Aaron Wan-Bissaka.
The hosts doubled their lead in the 27th minute when ex-Borussia Dortmund forward Jadon Sancho broke into the box from the left hand side, glided past James Tarkowski, before picking out the corner via a nick off Ben Mee's boot.
Wood wasted a golden opportunity to reduce the arrears following a tussle with Harry Maguire, but, having battled to get goal side of the England international, the striker's effort was blocked with Aaron Lennon, unmarked just yards to his left, in a much better position.
And the visitors were punished again. Hennessey turned McTominay's effort on to the woodwork only for Ronaldo to flick the ball into an empty net, scoring against the 121st team of his career in the meantime.
Sean Dyche's side handed themselves a lifeline before the break when Lennon intercepted possession from Wan-Bissaka, skipped past the challenge of McTominay, raced alongside Maguire and threaded the ball into the far corner.
Burnley kept the home side at bay after the break, with Ben Mee and James Tarkowski both heading wide from a couple of Ashley Westwood corners, though Hennessey produced a number of excellent saves to keep out McTominay's piledriver and Edison Cavani's close range effort.
Here are the player ratings.
1. Wayne Hennessey 6.5
Unsighted for United's first, left hopelessly exposed for the home side's second and hugely unfortunate for the third. Raced off his line to clear the danger at the feet of Ronaldo, made a stunning save to deny Cavani and flicked a McTominay piledriver over the top.
2. Matt Lowton 6
Caught out by Shaw's pass around the back when Ronaldo fluffed his lines and unable to nick the ball off the full back leading up to United's second. However, delivered a peach of a cross for Wood at 0-0 and reduced Sancho to anonymity.
3. Charlie Taylor 6
Kept the usually menacing Greenwood at arm's length, nipping any potential danger in the bud at the first opportunity. Tucked in well to support his centre backs, made a good block to deny Ronaldo in the first half and did his all to get up in support of Gudmundsson/McNeil.
4. James Tarkowski 6
Neither Ronaldo or Cavani posed the centre back too many problems in and around the penalty area, though he was beaten far too easily by Sancho when showing the ex-Borussia Dortmund star on to his right foot. Went close with a late header from Westwood's corner.