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.
5. Ben Mee 4.7
Left exposed by Harrison's defence-splitting pass for the away side's third goal and nowhere to be seen when Rodrigo was given time and space to net his second. Did the basics well when clearing from wide areas, but an afternoon to forget for the skipper. Photo: Catherine Ivill
6. Josh Brownhill 4.2
Covered plenty of ground, getting across to break up the play at times when Leeds threatened to counter, but the prominence of Harrison and Alioski made it difficult for him to roll the dice and get forward. Can't fault his effort. Photo: Julian Finney
7. Dwight McNeil 4.3
Came off the line, affecting the game in the middle of the park, protected the ball and able to get out of some sticky situations. However, looks in desperate need of a break with fatigue kicking in early in the second half. Photo: Oli Scarff
8. Ashley Westwood 5.3
Overrun in midfield when Leeds took control in the second half. Persevered in trying to get the strikers in behind, always provided an option for the player on the ball, and wasn't far away with a couple of well hit attempts from distance. Photo: Molly Darlington