The visitors kicked off attacking the River End goal, but it was the home side who had the best of the early exchanges.
Thereafter, there was little action in either penalty area with the game deadlocked in a crowded midfield.
Then, for the second time in the game, Norwich fell behind to a goal in the last minute of the first half.
Gareth Ainsworth beat Paul Gallacher from 16 yards with a low angled drive from the inside right position.
In the second half Robert Earnshaw finally managed to spring the offside trap but his angled drive flashed wide.
If the first half had lacked goalmouth action the second had plenty.
Gallacher did well to keep out Marc Nygaard's header and the ball was hacked off the line by Ryan Jarvis to deny Danny Shittu a chance to tuck in the rebound.
At the other end Paul Jones did well to keep out Darren Huckerby's shot before QPR made it 2-0 when Lee Cook waltzed through the City back line and beat Gallacher's despairing dive to the left with a well-struck left-foot shot from 16 yards.
Six minutes later Cook had another chance to test the City keeper with a free-kick but Gallacher made a comfortable save.
Ainsworth forced his keeper to make a save from a misdirected defensive header, then Huckerby did put a header past Jones when the Rangers keeper flapped at a right-wing cross from Andy Hughes.
Gallacher then made an excellent save to keep out a header from Nygaard before Earnshaw put Norwich level.
Jonatan Johansson, who had replaced Jarvis a few minutes earlier, put in a low cross from the right and Earnshaw finished at the far post with a right-foot shot from inside the six-yard box.
Gallacher pulled off another incredible save to deny Paul Furlong before Earnshaw secured victory in the 90th minute.
Paul McVeigh's deep cross from the left was headed back across the face of goal by Gary Doherty and Earnshaw headed the ball home from close range ahead of Dickson Etuhu.