Bidwell headed home Luke Freeman's left-wing corner with 15 minutes remaining to give the west London club their first third-round win without the need of a replay in 23 years.
Leeds had hit back after Aramide Oteh's 23rd-minute penalty put Rangers ahead at Loftus Road.
Young forward Oteh calmly converted from the spot after Bidwell had been brought down by Leif Davis.
However, the visitors were level less than two minutes later. Goalkeeper Matt Ingram, given a chance to impress having been kept out by Joe Lumley this season, fumbled Lewis Baker's free-kick and Aapo Halme pounced on the loose ball to equalise.
Both sides had chances after that, with Leeds goalkeeper Bailey Peacock-Farrell producing several saves – including two other Bidwell headers from Freeman corners – before the winning goal.
Prior to going ahead, QPR were denied by Peacock-Farrell on three occasions after almost going behind in the opening minute.
Tyler Roberts came within a whisker of putting Leeds ahead after just 30 seconds when his shot struck both posts.
Rangers responded to the early scare and Peacock-Farrell pushed away a shot from Oteh and Bidwell's header from the resulting Freeman corner.
Peacock-Farrell then produced a point-blank save to deny Bright Osayi-Samuel, who was put through by Ebere Eze's clever flick.
Peacock-Farrell was sent the wrong way by Oteh's penalty but his team were not behind for long as Halme levelled two minutes later.
He was, however, called into action again before the interval, gathering Osman Kakay's shot before seeing efforts from Eze and Oteh whizz wide.
Jordan Cousins almost restored QPR's lead in spectacular fashion midway through the second half.
The midfielder picked up a loose ball in midfield and evaded two challenges before unleashing a right-footed strike which the excellent Peacock-Farrell dived to his right to push on to the post.
Rangers continued to press and Darnell Furlong headed over from Freeman's looping cross, before Leeds threatened with Ingram atoning for his early blunder – and several other shaky moments – by palming away a shot from Ezgjan Alioski.
Peacock-Farrell then kept out Bidwell's far-post header from Freeman's right-wing corner, but the defender's persistence was rewarded when he nodded in Freeman's delivery from the opposite flank.
Freeman almost made it three when his late free-kick hit the post.