Mateusz Klich gave the Whites the lead just before the break with a beautiful effort from distance that nestled in the corner.
Marcelo Bielsa's side continued where they left off after the break and opened the Clarets up at will.
Jack Harrison doubled the away side's lead when flicking Ezgjan Alioski's wayward effort past Bailey Peacock-Farrell.
Rodrigo, who replaced Patrick Bamford just before the hour, ensured things would go from bad to worse for Burnley.
The Spanish striker clipped the ball over Peacock-Farrell after Harrison's pass split the home side's centre-halves in the 77th minute.
And the two combined again two minutes later as United's £27m club record signing took the ball around the goalkeeper before firing home.
Here are the RATINGS.
9. Jack Cork 4.5
Showed good strength on and off the ball in the middle of the park when the game was still competitive. Had Phillips' number in the first half, but the pendulum swung after the break. Another workmanlike display, though largely an uphill struggle. Photo: Gareth Copley
10. Chris Wood 4.2
Given very little change out of Llorente and Strujik and was replaced by Barnes just before the hour. Won a few flick-ons but couldn't make the ball stick for long enough when the hosts got the ball forward from the back. Photo: Peter Powell
11. Matej Vydra 5.2
Made a nuisance of himself and, once again, was an effective out-ball for the Clarets. Always a willing runner, getting something out of nothing when chasing seemingly lost causes, and unfortunate not to get a second half equaliser. Photo: Peter Powell
12. Ashley Barnes 3.5
Should have done better with a late header, but the game was dead and buried by that point. Gave Llorente and Strujik a bit more of a physical tussle in the final 25 minutes of the game and brought others into play. Photo: Lindsey Parnaby