Juan Mata's ninth goal of the season 18 minutes from time kept alive United's hopes of a top four finish and pushed Alex Neil's Norwich closer to the drop.
In a low key game with little excitement, Mata's solitary strike put the Reds to within a point of closest rivals Manchester City (who host Arsenal on Sunday) with a game in hand.
For the hosts however, it proved to be a fourth successive defeat- all without a goal- and after Sunderland's win over Chelsea, the East Anglian outfit now looked doomed to an immediate return to the Championship.
The visitors lost Anthony Martial to injury in the warm up as Ander Herrera stepped up and Matteo Darmian left the field with his foot in a brace to further compound the visitors injury worries.
In a largely uneventful and attritional contest, Cameron Jerome should have done better early on with a header whilst Chris Smalling also nodded over and Jesse Lingard fired weakly at Canaries keeper John Ruddy.
Memphis, one of four changes made by Louis van Gaal, then summed up the first half frustrations when he fired a free-kick straight into the wall when well placed 20 yards out.
Gary O'Neil sliced a speculative effort wide and Jerome misplaced a header before the breakthrough arrived at the other end with 18 minutes to play.
Antonio Valencia's pass over the top was misjudged by the back-pedalling Sebastian Bassong, Wayne Rooney collected the ball and advanced towards goal.
The United captain had to check inside Russell Martin and Bassong but did enough to lay the ball back for Mata, who arrived on cue to steer in a low left-footed finish.
Jerome fired off target and Wes Hoolahan went close, but substitute Morgan Schneiderlin should have made the points safe on 85 minutes.
The Frenchman blasted wide from close range having been released by Rooney, Memphis curled a free-kick narrowly past the post and Herrera lashed over the bar in stoppage time as United remained in control.
A match we never looked like losing and we made hard work of it, but it means we head to West Ham on Tuesday with all still to play for in the race for a top four place.
Overall team performance: 6/10
United Faithful Man of the Match: Juan Mata. Intelligent, bright and capped his performance as our best player with the decisive goal