Rangers had not won in a month and looked destined for defeat as Luton bombarded them in the opening 45 minutes.
But Mike Newell's men were made to pay for missed chances as Cook's second-half effort proved enough to win the game for Rangers.
Luton were the better side from the start and could have gone ahead in the first minute. Warren Feeney ran through on goal but his effort was palmed away by keeper Simon Royce.
The visitors continued to dominate and Kevin Nicholls' shot was deflected over the bar moments later.
QPR were limited to just two chances in the entire half. First Tommy Doherty shot wide on 11 minutes from outside the area and Paul Furlong crept offside when put through by Gareth Ainsworth just before half-time.
The best chance of the opening period came on 20 minutes and Ahmet Brkovic should have scored. Nicholls set him through with a perfect pass, but midfielder Brkovic hooked his effort wide.
Luton went in at the break hoping they would not be made to pay for missed chances, but five minutes into the second half QPR almost caught them out.
Cook's outswinging corner was met by the head of Ainsworth, but his header was brilliantly kept out by Luton keeper Marlon Beresford.
On 54 minutes Furlong tried his luck with a long-ranger which was easily kept out and soon after Rangers took the lead.
Chris Coyne brought down Furlong 20 yards out and Cook stepped forward to curl a low free-kick into the bottom corner of the net.
Rangers were well on top by now as Luton had lost their rhythm and on 66 minutes Furlong should have doubled the lead for Ian Holloway's men.
Richard Langley was brought on as a substitute for Ainsworth, and made an immediate impact. He raced on to a loose ball to tee up Furlong eight yards from goal, but the striker was off balance and scuffed his shot over the bar.
Ten minutes from time Rangers were reduced for ten men. Doherty got into a scuffle with Luton's Coyne and appeared to take a swing at him.
Referee Mike Thorpe wasted no time in reaching for his red card to send Doherty packing, but Rangers held on to claim the win.