Martin fired City ahead in the 10th minute - his first goal at Carrow Road - to secure a fifth win in six games in front of a crowd of more than 26,000.
QPR had defender Matthew Connolly sent off after 51 minutes for a professional foul on City captain Grant Holt.
But, after the Canaries wasted several chances to kill the game off, substitute Patrick Agyemang headed against a post with 16 minutes left before being denied by goalkeeper John Ruddy at the death.
Norwich, though, held on to post their first clean sheet in 11 games and climb to third in the table.
Source: PA
Source: PA