Juan Mata scored the winner for the Reds in the 72nd minute after a mistake from Sebastian Bassong let the Spaniard into score. The bad news though for United was that they suffered two more injuries in the form of Anthony Martial and Matteo Darmian.
Martial injured moments before kick-off
Moments before kick-off Martial was injured in the warm-up and was replaced by Ander Herrera in the starting lineup to give Louis van Gaal a further headache.
The Canaries began the game on the front foot and should have gone in front when Nathan Redmond played a lovely cross into the box which found Cameron Jerome, but the striker saw his header brilliantly saved by David de Gea.
Darmian suffers bad injury early on
Darmian then went off injured 16 minutes into the game with what looked like a bad injury after going off on a stretcher.
United finally created their first chance of the game on the half-hour mark when Chris Smalling headed a wonderful cross from Mata over the bar. The defender got above his man in the box and was only four yards out, but somehow he put his header over the bar.
The rest of half saw both teams try to score, but both just didn't have the quality to create anything with a lot of misplaced passes which was frustrating for van Gaal to see his players unable to create much.
United much better at the start of the second half
The Reds came out at the beginning of the second half with more intent and almost took the lead when a lovely pass from Mata found Herrera in space in the box. The midfielder took the pass on the volley but put the shot into the side netting.
The Reds continued to have most of the ball but on the counter, Norwich almost scored but Dieumerci Mbokani mistimed his header unmarked in the box. Wes Hoolahan produced the cross for the striker who will be disappointed he didn't do better.
Mata scores winner to keep United in the race for a top-four spot
United though finally made their second half dominance of the ball count when Mata opened the scoring for the Reds.Bassong made an awful mistake for the goal when he misjudged a long ball over the top and Wayne Rooney held the ball up before passing to Mata and the Spaniard stroked the ball home to give the Reds an important lead.
Morgan Schneiderlin almost made it two for United late on when Rooney played the midfielder in on goal after holding the ball up well. Schneiderlin though showed that he doesn't score many goals as he put his shot wide of the goal.
The second half ended to give the Reds a crucial win in the race for the top-four with two games remaining. United. United now move onto Tuesday evening where they will play West Ham United at the Boleyn Ground in a massive game for both teams.