Birmingham had lost 6-1 at Hull City prior to the international break, but looked like a different side on Saturday (AEDT).
Che Adams' excellent individual goal after 19 minutes, his first league strike of the season, proved decisive to give Cotterill, who officially took charge last week, an impressive start that moves the Blues out of the bottom three.
Leader Cardiff failed to produce a shot on target, meaning Wolves can go top of the table with at least a draw against Aston Villa on Sunday, with Sheffield United also in striking distance.
"I'm really pleased," said Cotterill. "I just thanked the lads and all the staff for their efforts over the last two weeks. They have been first class."
Elsewhere in the Championship, Bristol City missed the chance to go second in the table as it was held to a 0-0 draw at home to Burton Albion, which moves out of the bottom three after a battling defensive effort.