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.
1. Bailey Peacock-Farrell 4.2
You have to have some sympathy for the Northern Irish stopper, who has now shipped 14 goals in four appearances in the PL. Wasn't necessarily given the level of protection required to negate a potent and dynamic Leeds United. Photo: Alex Livesey
2. Matt Lowton 4.3
Started well, getting out to Harrison and Alioski quickly to stunt their flow, but the visitors soon got into their stride. Couldn't contend with the speed of the away side's counters in the second half with Leeds quick to get the ball out into the wide areas. Photo: Clive Brunskill
3. Charlie Taylor 5.8
The best of a bad bunch. Although Raphinha got the better of him on occasions, particularly in the first half, he did well up against the tricky Brazilian. Carried the ball forward remarkably well and, at times, felt like Burnley's only outlet. Photo: Clive Brunskill
4. James Tarkowski 4.6
An inconsistent display from the Burnley centre back. Made a couple of good blocks and interceptions and had the better of Bamford. However, showed limitations for a number of goals conceded and missed a late chance from close range.. Photo: Clive Brunskill