Cook was wanted by Manchester City, Spurs and Everton in the January transfer window and he was a nuisance to the Clarets defence, scoring the first with a low drive and setting up Dexter Blackstock's second with an inch-perfect corner.
The away team had the first chance of the game when a clever header from Ade Akinbyi sent Alan Mahon through. The Burnley midfielder was shaping to shoot when Sampsa Timoska made a last-ditch challenge.
But with Burnley pushing forward, it was QPR who took the lead on 12 minutes with a crisp counter-attack. James O'Connor lost the ball to Adam Bolder in the home side's half, and the new signing found himself with space to run into.
He pushed across the halfway line and played the ball into the path of the on-rushing Cook, who held off John McGreal and fired a left-footed shot into the bottom right from just inside the box.
It only took six minutes for the Clarets to draw level. Another Akinbiyi knockdown allowed Joey Gudjonsson to cross from the right. Chris McCann ghosted into the box unmarked and planted his header past Simon Royce from 12 yards out.
Burnley dominated possession for much of the half, but created few chances, and the Hoops went into half-time looking the more likely scorers.
After a long build-up, Timoska found Blackstock with a cross from the left, but the striker could only head wide.
The home side started the second period as they finished the first, with Gareth Ainsworth firing wide after Cook had jinked past Steve Foster and lofted a dangerous cross into the centre.
QPR capitalised on a dominant spell with a goal on 54 minutes, when Blackstock rose brilliantly above the Burnley defence to fire a header into the top right from Cook's outswinging corner.
Cook began to stamp his authority on the game and he volleyed just over on 57 minutes from a floated Blackstock cross.
QPR deservedly made it 3-1 after 72 minutes, when Steve Lomas headed into the bottom right from a Martin Rowlands corner on the right.