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.
13. Jay Rodriguez 3.4
Worked hard across the front, looking to win the ball in key areas in the final third, but couldn't quite get into the game. Should have done better when a chance presented itself after his introduction, with Meslier saving comfortably. Photo: Clive Brunskill
14. Johann Berg Gudmundsson 3.7
Looked lively after replacing Dwight McNeil with 20 minutes remaining. Involved in some of the home side's finer moments in the closing stages and will have been disappointed not to have beaten Meslier when getting on the end of Westwood's pass. Photo: Clive Brunskill