A three-goal blitz in the first half was enough to give the Tigers a tremendous win at St Andrews, their first there for 42 years.
Sone Aluko stroked home two clinical finishes in the opening 25 minutes, then took the corner that James Chester headed in on 33.
Birmingham fought back with a Ravel Morrison strike and an inevitable goal from Marlon King either side of the break but City held on to remain fourth in the table.
Home games against Burnley and leaders Crystal Palace next for the Tigers, who look in excellent shape for both games. Match report later.