The Canaries collected a valuable away victory carved out by a superb first-half goal from the outstanding Alex Pritchard and a second from Josh Murphy.
It was a real dogfight and a dreadful result for a Birmingham side which has now won only two games in 13 outings under manager Steve Cotterill and scored just five goals.
Birmingham, severely handicapped by their lack of strikers, went into the game with only one win in 11 games while Norwich had just one victory in 10 outings which demonstrated why it was such a tense match.
With James Maddison as their playmaker, Norwich were bright and breezy in the opening exchanges but Birmingham battled their way back into contention with some determined moves but failed to create any really serious goalscoring opportunities.
Lukas Jutkiewicz, drafted into the side in place of the suspended Sam Gallagher, failed to make much impression against a Norwich defence which effectively funnelled back at the sight of danger.
Birmingham’s best chance of breaking the deadlock occurred when Jonathan Grounds went on a run finishing with a curling shot towards the far top corner of the net forcing Angus Gunn to make a fingertip save.
Norwich, without the services of Cameron Jerome who had scored three goals in his last two starts against Birmingham, had to mount a defensive blockade with Timm Kiose playing a key role.
The industry of Craig Gardner emerged as an important feature of Birmingham’s play, but the performance of their expensive signing, Jota, left a great deal to be desired.
Birmingham’s commitment to attack left them vulnerable as was demonstrated when Norwich took a 31st-minute lead.
Maddison provided a low pass to the right from a smart set-piece routine, enabling Pritchard to run onto the ball and score with a powerful angled shot.
The goal immediately resulted in Norwich adopting an even more defensive shape in a bid to retain their slender advantage, but they still looked more dangerous than the hosts as Nelson Oliveira was denied a second goal after the break by another smart save from David Stockdale.
Norwich settled the game when Pritchard astutely set up Murphy with a through-ball, and Murphy took the ball around Stockdale to score.
The Birmingham goalkeeper prevented a heavier defeat with two outstanding saves, but there was no way back for Blues.