Anderson's header broke the deadlock on 68 minutes, to the home fans' relief, and substitute Danny Welbeck clinched the three points with his third goal of the week at Old Trafford.
United looked flat in the first half and struggled to make any superiority count against hard-working and well-drilled opponents.
Although Wayne Rooney had an early shot straight at John Ruddy, it was City who started to make the running and create goalscoring opportunities. Anders Lindegaard, starting ahead of David De Gea, allowed Bradley Johnson a sight of goal when he sent a clearance straight to the former Leeds midfielder but, to the Reds keeper's relief, the shot was weak.
Only a vital clearance from Phil Jones snuffed out the danger sparked by a run from Elliott Bennett down the right and Wes Hoolahan also managed a couple of efforts from outside the box.
After Javier Hernandez was blocked by Marc Tierney when trying to reach a Rooney cross, the Canaries again showed they were not going to be overawed when Steve Morison beat Jonny Evans with his chest control and pulled his finish wide.
Gradually, following the half-hour mark, United started to ask more questions of the yellow shirts at the back and Tierney volleyed away when Darren Fletcher flicked a Nani corner goalwards. Rooney climbed well to divert