Two errors from recently-capped England international Jack Butland gifted Wednesday a two-goal lead.
Spanish striker Rodri, who joined the Owls on loan from Barcelona earlier this week, then doubled the hosts' advantage by firing through Butland's legs from inside the box.
Towering forward Nikola Zigic was introduced to the action as Lee Clark sent his side out in search of a second-half comeback.
The Serbian was responsible for pulling the Blues back into the match as he fired in at the far post following Nathan Redmond's inviting cross.
Birmingham pushed for an equaliser but instead Wednesday restored their two-goal cushion with a Jermaine Johnson goal on the counter attack.
Marlon King netted a stoppage-time penalty for the Blues but it proved to be the last kick of the match, and too little, too late for the West Midlands outfit.