QPR kicked off attacking the River End goal and only a diving one-handed save by David Marshall prevented Angelo Balanta from giving them an early lead.
However inside the first seven minutes the visitors were a player and a goal down. Ched Evans was fouled on the edge of the penalty area and Damion Stewart was sent off for the foul.
Evans then scored with a firm right-foot volley which beat Lee Camp high to his left when Rangers failed to deal with Darren Huckerby's cross.
Rangers responded by making a double substitution, Hogan Ephraim and Gareth Ainsworth making way for Zesh Rehman and Mikele Leigertwood.
It was the 19th minute before City threatened again, but Mark Fotheringham's shot was blocked and Matty Pattison fired wide.
After 23 minutes Huckerby had a chance, but he too curled his shot wide and three minutes later Evans did likewise.
On the half hour Ryan Bertrand's shot came back off a post and it was first-half stoppage time before Norwich had another chance, when Jamie Cureton fired over from 18 yards.
After Camp had done well to keep out shots from Evans and Cureton, 11 minutes into the second half Fotheringham's right-foot shot from 18 yards took a deflection and beat the QPR keeper low to his left.
In the 66th minute, with Camp well out of position, Fotheringham missed a chance to kill off the game, as his long-range shot drifted wide.
Cureton put in a good cross from the right but it eluded everyone.
Evans fired wide and Kieran Gibbs' shot was saved by Camp. However, the keeper went walkabout once too often and substitute Darel Russell found the back of the net with a long-range left-foot shot to ensure victory for the Canaries.