A run of four games without a victory had seen Norwich slip out of the automatic promotion places, but Daniel Farke's side got back to winning ways in emphatic fashion thanks to a first-half blitz.
Top scorer Teemu Pukki notched his 16th of the season from Jamal Lewis's low cross, but Birmingham hit back 78 seconds later as Che Adams converted Jota's sublime defence-splitting pass.
Yet Mario Vrancic and Tom Trybull scored in quick succession before the half-hour mark, with Birmingham's defence all at sea in front of Lee Camp.
Vrancic was allowed too much space to fire into the bottom corner from 25 yards, and Trybull was left unmarked three minutes later to head home Emiliano Buendia's left-wing corner.
Chances were harder to come by after the break, with Camp making a good save to deny Todd Cantwell before a last-ditch tackle from Gary Gardner stopped Pukki grabbing a second.
A weak Adams effort from long range was the best the visitor could muster, but Tim Krul saved comfortably.
Birmingham has gone four games without a win after the defeat and is in danger of dropping into the bottom half of the table should results go against it on Sunday (AEDT).