The Potters went ahead early on through Jon Walters, but Heidar Helguson equalised in the 22nd minute and Luke Young then struck at the end of a flowing move before the interval.
Helguson made it 3-1 in the 54th minute - his fifth goal in five games - from Joey Barton's cross, and although Ryan Shawcross's header 10 minutes later reduced the deficit, the hosts were unable to prevent a fourth straight defeat in the league.
QPR, meanwhile, can celebrate a third away win of the season in an impressive return to the top flight so far.
Stoke boss Tony Pulis made five changes, the most notable seeing goalkeeper Thomas Sorensen replacing Asmir Begovic, who had been at fault for two goals in the 5-0 defeat at Bolton on November 6.
Fit-again pair Matthew Etherington and skipper Shawcross were restored to the side, while Danny Higginbotham also made his first league start of the campaign.
QPR made just change from the 3-2 loss to Manchester City, with striker Jay Bothroyd not even named as a substitute and former Stoke defender Clint Hill the man promoted to the team.
The home side had appeals for a penalty waved away by referee Mike Jones in the seventh minute after Jermaine Pennant went down in the box under the attentions of QPR captain Barton.
Seconds later Stoke took the lead, though, as QPR lost possession and the ball came to Crouch, who slid in a pass for Walters to finish.
At the other end Alejandro Faurlin hit a tame shot at Sorensen before Crouch came close to doubling Stoke's advantage, pouncing on a moment of hesitancy between Anton Ferdinand and Paddy Kenny but seeing the Rangers goalkeeper block his first attempt and then his second cleared away by Hill.
Kenny was having a torrid time, fumbling an Etherington free-kick before smothering the ball at the second attempt.
But against the run of play QPR then drew level, Helguson being allowed the space to head past Sorensen from Armand Traore's cross.
Stoke kept pressing, with Robert Huth heading a Pennant corner over the bar and Etherington striking wide after a mazy dribble into the area.
Rory Delap sent a shot off target and it looked as if QPR would do well to hang on until half-time.
But they made it 2-1 in the 44th minute, good link-up play leading to Jamie Mackie's cut-back for Young, who lashed the ball in.
Stoke came out for the second half looking to make a swift response, but within nine minutes of the restart QPR had another goal.
Barton crossed after the hosts had failed to properly clear a corner and Helguson was on hand to prod the ball in from close range.
Helguson then thought he had completed his hat-trick when Shaun Wright-Phillips hit the post and the Icelander scrambled the rebound over the line, but the assistant referee's flag was raised for offside.
Stoke rallied and pulled one back in the 64th minute when Crouch flicked a corner to Shawcross, who headed beyond Kenny.
Hoping to seize the initiative, Pulis refreshed his attack by bringing on Kenwyne Jones and Ricardo Fuller - the Jamaican's first appearance since sustaining an Achilles injury in April - for Crouch and Pennant.
Huth fired an ambitious attempt over the bar from the corner of the box with around 20 minutes to go as the Potters went in search of a leveller.
The German was then felled in the area by Barton's tackle, but once again the referee waved play on and Stoke's frustration was apparent when Delap picked up a booking for a wild lunge on Barton and one of Stoke's backroom staff appeared to be sent off.
Stoke went close late on with substitute Glenn Whelan sending a drive whistling wide and Kenwyne Jones heading over in stoppage time, but QPR survived to claim all three points.
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