Chances were few and far between in the opening 45 minutes but Ashley Young produced a fine save from John Ruddy following a United corner, while Chris Smalling did well to intercept a low Robert Snodgrass cross that was destined for the boot of Grant Holt at the near post.
Norwich started strongly after the break and took a deserved lead through Pilkington on the hour mark. Wes Hoolahan, who was impressive throughout, linked up well with Javier Garrido and the latter's tempting cross found Pilkington, who beat Anders Lindegaard with a glancing header.
United struggled to break down a resolute Norwich side, who defended in numbers after taking the lead, but Danny Welbeck managed to get a headed effort on target following a Robin van Persie corner on 76 minutes, although it was a simple save for Ruddy.
Substitute Johnny Howson wasted a glorious opportunity to seal the win for Norwich in the dying stages as he shot wide of the target when one-on-one with Lindegaard.
United twice came close in injury time but Ruddy was alert to the danger when Sebastian Bassong inadvertently sent the ball goalwards with his head from a Van Persie free-kick, the Canaries stopper then denied Anderson and managed to smother the ball as Paul Scholes tried to reach the rebound.