However, the visitors had a great opportunity to level the scores eight minutes later when Djibril Cisse was fouled by Sebastien Bassong and the referee awarded a penalty to the away side.
Cisse's poor effort was saved by John Ruddy but Bobby Zamora reacted well to stroke the ball past the helpless goalkeeper.
Norwich continued to be the better side after the equaliser and Green kept his side in the game with a string of impressive saves from Anthony Pilkington and Bradley Johnson before half-time.
Chris Hughton's team continued to control possession at the start of the second half and the hosts were infuriated when Green appeared to handle the ball outside the area on the hour mark. However, the referee refused to punish the England international.
Pilkington troubled QPR throughout the match and his dangerous play on the wing almost produced a winner but Russell Martin could only head the midfielder's cross against the woodwork.
Neither side produced any clear chances in the final moments, with both sides happy to get their first point of the season.