Paul McVeigh and Dean Ashton hit the target before an amazing blunder by David Unsworth gifted the points to the Canaries and killed off any hopes of a fightback.
Norwich should have snatched the lead in the fourth minute. McVeigh skipped past a tackle to pick out Ashton but the big striker pulled his left-foot shot wide from ten yards.
United finally broke the deadlock in the 22nd minute. Alan Quinn found Phil Jagielka with a corner from the right and the midfielder jumped at the near post to head firmly past Robert Green.
Norwich were back on level terms in the 33rd minute with a classy strike. Ashton's long ball across field found Carl Robinson and he headed to the waiting McVeigh who beat Paddy Kenny with a low drive from 16 yards.
Michael Tonge almost found a way through the City defence, cutting in from the left wing to finish with a low shot, the ball flashing across the face of the goal but wide of the target.
Norwich snatched the lead on 61 minutes, full-back Adam Drury finding the £8million-rated Ashton on the left and as United desperately looked for an offside flag, he coolly slotted the ball wide of Kenny from 12 yards.
Unsworth's mistake came in the 70th minute. Kenny had a rush of blood as he charged out of the area in a race for the ball with Darren Huckerby.
The Norwich winger side-stepped the keeper and picked out Ashton whose shot was drifting wide before the big full-back intercepted and headed the ball into an empty net.