Juan Mata punished Sebastien Bassong's error as United moved within one point of neighbours Manchester City, who play Arsenal on Sunday.
Norwich had 18 minutes to do what they had not managed to in their three previous games - score - but once more they failed to do so.
The Canaries' fate could be decided as soon as Wednesday, when they host Watford.
Suddenly everything is looking more positive for Louis van Gaal.
The United manager, whose position has been under scrutiny for much of the season, could yet finish the season in the top four and by lifting the FA Cup.
Norwich won 2-1 at Old Trafford on December 19 and were aiming to inflict United's first home-and-away loss to a newly-promoted side in the Premier League era.
The hosts began with intent, a strong early spell coming between injury blows for the visitors.
Anthony Martial was ruled out during the warm up and, after 16 minutes, Matteo Darmian departed on a stretcher with an apparent ankle injury.
The Canaries played at high intensity and with purpose.
Robbie Brady ran by Carrick, with Mata trailing in his wake, but stumbled.
Bassong had a shot blocked and then Nathan Redmond crossed for Cameron Jerome in the fourth minute.
That should have been 1-0, but the striker's header was straight at David de Gea.
Jerome's most recent goal had been at Old Trafford and the subsequent 16-game scoreless streak has seen his confidence dwindle.
Any momentum Norwich had was lost after Darmian fouled Ivo Pinto, resulting in a pause for treatment.
United were happy to let the Canaries have the ball and play on the break
From one Mata corner, Chris Smalling headed over.
Jerome encapsulated Norwich's lack of quality, repeatedly relinquishing possession.
Bar the busy Wes Hoolahan, the hosts barely knew what to do.
Jerome made one surge down the right, skipped by Marcos Rojo only to limply fall following the effort of the run.
The striker showed more intent in splitting up Rojo and Russell Martin after the Norwich skipper reacted to a swinging arm which made little or no connection as the pair grappled following a corner.
A Hoolahan cross was met by Dieumerci Mbokani, on for Brady, but the striker's header merely looped into the air and was claimed by De Gea.
Norwich fared little better in the second half and the sense of inevitability was growing among home supporters who had seen it all before.
An Antonio Valencia ball down the right channel evaded Bassong, who committed to meeting it, and allowed Wayne Rooney in.
Rooney, up front in place of Martial, could not find space to shoot but had the presence of mind to roll the ball to Mata to tuck in beyond John Ruddy.
Norwich's win over Newcastle on April 2 - a result which boosted their survival hopes - was the last time the Canaries had scored.
Crystal Palace, Sunderland and Arsenal had kept cleansheets in the meantime and Norwich were unable to trouble United's bid for one.
It was United who finished looking most likely to score.
Substitute Morgan Schneiderlin scuffed a shot wide across goal after a fine Rooney pass, Memphis Depay curled a free-kick just wide and Herrera blasted over.
TWEET OF THE MATCH
"Heartfelt gesture from Mata staying behind to have a picture with a young disabled #mufc fan on the touchline
Top man." - MEN football writer Samuel Luckhurst (@samuelluckhurst) comments on Juan Mata taking a picture with a disabled fan after their win.
PLAYER RATINGS
Norwich
John Ruddy: 6 (out of 10)
Ivo Pinto: 5
Russell Martin: 5
Sebastien Bassong: 4
Martin Olsson: 5
Jonny Howson: 6
Gary O'Neil: 6
Nathan Redmond: 7
Cameron Jerome: 5
Wes Hoolahan: 7
Robbie Brady: 5
Substitutes
Dieumerci Mbokani (for Brady, 63): 5
Graham Dorrans (for Howson, 78): 5
Patrick Bamford (for Jerome, 78): 5
Manchester United
David de Gea: 6
Antonio Valencia: 6
Chris Smalling: 7
Marcos Rojo: 5
Matteo Darmian: 5
Michael Carrick: 5
Jesse Lingard: 5
Ander Herrera: 6
Juan Mata: 7
Wayne Rooney: 7
Memphis Depay: 6
Substitutes
Cameron Borthwick-Jackson (for Darmian, 16): 6
Morgan Schneiderlin (for Lingard, 78): 5
Timothy Fonsu-Mensah (for Carrick, 88): 5
STAR PLAYER
Smalling
Composed amid the mediocrity, holding together United's reshuffled defence.
MOMENT OF THE MATCH
Mata's goal
It was, in many ways, Norwich's season in microcosm
A defensive error (from Bassong), a frantic bid to cover (when Rooney went through) and then a failure to pick up the key man (Mata).
VIEW FROM THE BENCH
Alex Neil named an unchanged starting XI following the loss at Arsenal, while Van Gaal made three changes for the visitors, with Darmian, Mata and Depay in for Daley Blind, Marcus Rashford and the suspended Marouane Fellaini
A fourth change followed when Martial was injured in the warm-up and Ander Herrera drafted in to start
Neil persisted with a struggling Jerome, throwing on Mbokani to join him up front, but the change of shape did not improve Norwich's chances.
MOAN OF THE MATCH
Norwich failed to score for the fourth straight Premier League game
They are hardly showing that they are deserving of staying in the top flight.
WHO'S UP NEXT
Norwich v Watford (Barclays Premier League, Wednesday, May 11)
West Ham v Manchester United (Barclays Premier League, Tuesday, May 10)
Van Gaal said on BT Sport: "We're still in business, the players were very tired from last week but we won.
"We keep pressure on our competitors and we hope we can achieve a Champions League place because we want to play at the highest level possible but it is still very tough because we have to play away at West Ham."
Marcus Rashford and Daley Blind were rested for the game as United suffered injuries to Martial and Darmian but believes he made the right decision to rest both players.
"I stick with my decision and they needed rest, you see the players, they also need rest but I cannot give everybody rest so that's the problem.
"We have to see the reaction (to the injuries) tomorrow
Darmian got kicked in the ankle and Martial felt something during the warm-up."
Canaries boss Neil felt the game summed up how their season has fared.
"I think there were many opportunities in the game, one fell to Cameron Jerome in the first half
But when you gift goals and make errors then it's never going to be enough," he told BT Sport.
"I saw enough commitment, but ultimately it's about not making stupid errors and scoring goals if you get an opportunity
It's been the story of our season."
Source : PA
Source: PA