Bright Osayi-Samuel won and scored a late penalty to earn QPR a point against the Championship leaders.
But the winger then missed a golden chance to end Rangers’ winless run when he lifted the ball over from two yards out after being teed up by Albert Adomah’s low cross.
Teemu Pukki netted with a penalty with 16 minutes remaining after Dom Ball was adjudged to have fouled Todd Cantwell in the area.
Contact with Cantwell was minimal and Rangers, who had defended well, were furious with the decision.
But the visitors were themselves awarded a controversial spot-kick when Osayi-Samuel went down underrate similarly little contact from Christoph Zimmerman, and then thumped home.
Norwich had been dealt a blow when goalkeeper Michael McGovern had to be replaced at the interval by teenager Daniel Barden after suffering what looked like a hamstring injury in the final minute of the first half.
Barden was making his league debut but Rangers – now without a win in nine games – were unable to test the youngster until he denied them an equaliser by keeping out Ball’s header from Ilias Chair’s corner.
QPR were under sustained pressure before the opening goal and keeper Seny Dieng produced saves to deny Pukki and Emiliano Buendia, while Jacob Lungi Sorensen’s goal-bound shot was cleared by Rob Dickie.
There seemed no way back for Rangers after the deadlock was broken but they were handed a lifeline and Osayi-Samuel took full advantage – but was unable to do so again in the final stages.
QPR: Dieng; Dickie, Cameron, Barbet; Kane, Ball, Carroll, Hamalainen; Osayi-Samuel, Dykes (Bonne 70), Chair (Adomah 70).
Subs not used: Kelly, Masterson, Bettache, Thomas, Kelman, Willock, Duke-McKenna.