Brazilian nods in from a corner and young striker seals the game late-on, but the Premier League champions were a touch fortunate to take the points as Canaries wasted good chances
The Premier League newboys defended solidly for much of the game, and caused home side plenty of trouble on the counter attack, both early on in the game and in a period midway through the second half as the match was finely poised.
In truth, the Canaries probably should have taken all three points back to the east of England, but their finishing in front of goal was far too wasteful at a ground where mistakes go punished more frequently than most.
But United came out on top after the Brazilian midfielder's goal from a corner was compounded late on with a breakaway goal finished off by Welbeck.
For all of United's strength in attack, they almost came unstuck early on in defence when Patrice Evra was left flummoxed by a quick one-two from Elliot Bennett, who stole in behind and would have found an unmarked Steve Morison at the far post but for the intervention of Phil Jones.
As the first-half progressed Norwich looked fairly assured, though United were getting more and more time on the ball and looking to find gaps in the Canaries' defence with Nani, Wayne Rooney and Anderson the playmakers in chief, though they were failing to find a way deal with Norwich's defence.
Chances were certainly at a premium in the first half though. The closest the home side came to opening the scoring were at corners, with Darren Fletcher flicking the ball at the near post to be cleared, and Rooney missing the target when the ball bounced off his shoulder and over.
Elsewhere there was a shot from Nani that flashed wide from distance on the stroke of half-time, though Norwich's resolute defending and counter-attacking were serving them very well as they went into the break all square.
Norwich could perhaps be worried with the amount of time United were spending on the ball, with no real outlet up front to hold onto the ball.
The second half began as the first had finished, with United on the attack. Evra overlapped Nani on the left before putting in a dangerous cross, and Anderson dragged an effort wide after being well found by Rooney.
But in the 52nd minute, Norwich almost took the lead as Morison held off a challenge from Jonny Evans before charging goalwards. As he got towards the byline, Jones made a fabulous effort to get across and block the attempted cross, leaving an unmarked striker at the far post while he went.
Two minutes later and Anders Lindegaard was finally tested, as Anthony Pilkington cut inside of Phil Jones to shoot, but a deflection from a sliding Jonny Evans pushed the ball up to a fairly comfortable height, as he gathered the ball at the second attempt with help from Evra.
The game was becoming far more open and with half an hour to play Wes Hoolahan hit a shot from distance that flew close to goal, though Lindegaard had it covered.
The United players' frustration was clear to see, and Sir Alex Ferguson made the first move of the match, with Ryan Giggs and Danny Welbeck replacing Nani and Chicharito, who had both been rather disappointing.
And almost straight after the substitution, Norwich really should have been in front.
A quick break from the Canaries led to Pilkington chasing down Antonio Valenica in the right-back position, and the Ecuadorian player made a hash of it, gifting the ball to the winger, who bore down on goal, only to slot it wide as the Norwich bench seemed ready to explode.
And only minutes later, they were made to pay a heavy, heavy price as United found the lead through Anderson.
A Giggs corner was headed back in at the far-post and again in the centre by Rooney, and the Brazilian rose unmarked to nod the ball past John Ruddy and two defenders on the goal-line.
It was a touch harsh on the visitors who had defended soundly for over an hour, but no more than United deserved for their barrage on the Norwich goal.
With 15 minutes to go, Pilkington came as close as he ever will to scoring at Old Trafford without finding the net.
His driven shot after a corner took a deflection off Anderson and bounced high off the inside of the post and back down into the arms of the grateful Lindegaard.
It truly was a 'heart in your mouth' moment for every United fan inside Old Trafford, and sparked another change from Ferguson, who brought on Rio Ferdinand for Anderson which meant shifting the exposed Valencia up the pitch and moving Jones out to right-back.
It almost worked a treat as Jones went on a barnstorming run down the right to be found by Rooney, and his cross evaded the outstretched leg of Welbeck by mere inches.
Rooney was the next player to go close with 10 minutes to go, taking on the Norwich defence and hitting a shot just wide of the far post from a tight angle following a beautiful sweeping pass from Giggs.
And as Norwich pushed more men forwards in search of an equaliser, it was almost inevitable that the champions would find gaps in the defence.
With two minutes to go, United broke through Rooney, and his pass put Ji-Sung Park clear, before the Korean unselfishly squared for Welbeck to apply the finishing touch.
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