The visitors carved out virtually all the best chances, with Anthony Pilkington hitting a post and rolling a shot wide after finding himself clean through.
Norwich were made to pay when Anderson powered home a close-range header midway through the second period before Danny Welbeck scored his third goal in a week, sliding in to finish off Park Ji-sung's cut-back.
United boss Sir Alex Ferguson spent a good proportion of the build-up extolling the virtues of his Norwich counterpart Paul Lambert, and explaining why Scots are such a significant presence in the top flight.
Attention to detail clearly plays a huge part and Lambert had done his homework.
Norwich did attack when they could and they were the better team for the first 20 minutes, although their best work was done around their own penalty area - and it did not involve the ball.
Correctly assuming their opponents would dominate possession for long spells, the Canaries remained rock-solid in retaining two banks of four, which United found impossible to penetrate.
Returning striker Wayne Rooney unleashed a long-range effort, which Canaries goalkeeper John Ruddy dealt with easily and Marc Tierney was on hand to clear a near-post flick by Darren Fletcher from Nani's corner.
Other than that, the hosts were repeatedly frustrated, the unexplained absence of Ashley Young denying them a slippery customer on both flanks on a day when Nani was not at his best.
Certainly Norwich would not have expected to reach the interval in such comfort, the entertainment value so low that watching comedian - and Canaries supporter - Stephen Fry seemed to nod off in the directors' box.
Fry would have had plenty to cheer about if Elliott Bennett had been able to pick out Steve Morison after surging to the byline, the Norwich striker would have had a tap-in.
Instead, Bennett drilled his cross straight at Phil Jones and the hosts breathed a sigh of relief.
United survived two more gilt-edged Norwich chances immediately after the break too.
Jones has already caught the eye this season but the 19-year-old needed to show maturity way beyond his years after Morison had given Evans the slip. Knowing Pilkington was completely on his own behind him, the teenager held his ground, denying an early cross, before lunging in on Morison and coming away with the ball.
It was a textbook piece of defending, and a further evidence of why Ferguson felt so comfortable leaving Rio Ferdinand on the bench.
A minute later though, Jones was beaten by Pilkington, who stepped inside the former Blackburn man only for his shot to strike Evans, allowing goalkeeper Anders Lindegaard to make the save.
Wes Hoolahan was not that far away with a dipping long-range effort as Norwich, who had scored in 11 of their last 12 away games, pushed forward with relish.
United's good fortunate was seemingly inexhaustible. When Pilkington seized on Antonio Valencia's miscontrol, the striker was left with only Lindegaard to beat.
He steadied himself, just as Lambert would have wished, but amazingly rolled his shot wide of the far post, to the disbelief of the visiting contingent at that end of the ground.
Having passed up so many chances, Norwich must have feared what was about to follow.
Substitute Ryan Giggs floated over a corner, which was met by Jones and flicked on by Rooney, to Anderson, who powered home a header from close range.
Still, Norwich refused to let their heads drop and came agonisingly close to a leveller as Pilkington's shot deflected off Anderson, onto the post and back into the grateful arms of Lindegaard, who had been helpless.
Danny Welbeck then slid in but failed by inches to touch home Jones' cross.
Norwich continued to threaten but after Morison's goalbound effort was turned away by Ferdinand, Welbeck sealed the win with his third goal in a week before Rooney chipped narrowly over.
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