However, with England striker Wayne Rooney missing because of tonsillitis, the Red Devils produced a lacklustre display as the Canaries - reborn since the international break - produced another composed display to secure a third win in five Premier League games with a fine header from Anthony Pilkington on the hour.
Norwich, though, showed some life in attack when Wes Hoolahan floated a ball over from the left, which Snodgrass diverted just over the angle with a backward header.
Norwich continued to grow in confidence and it needed a brilliant saving tackle from Chris Smalling to prevent captain Grant Holt from tapping in Robert Snodgrass' low cross after the Scotland winger got past Rio Ferdinand into the right side of the United penalty area on 37 minutes.
Norwich continued to press and finally were rewarded on the hour with a goal of real quality. Javier Garrido's perfectly weighed cross was glanced over Anders Lindegaard into the top corner by Pilkington, who was once a player in the United youth system.
United upped the tempo, as Danny Welbeck's looping header was deflected behind for a corner and it continued to be one-way traffic as the Canaries defended resolutely.
There were four minutes of stoppage time and John Ruddy produced a brilliant saved to turn the ball over after it was deflected off Norwich defender Sebastien Bassong.