The Black Cats were two goals up inside an hour thanks to Darren Bent's penalty and an own goal from Stephen Carr.
They lost skipper Cattermole to a second yellow card for a foul on Lee Bowyer just before half-time.
And Blues defenders Scott Dann and Liam Ridgewell went on to score in the final 12 minutes to earn the visitors a point.
Bruce said Cattermole was initially booked for dissent, but the Sunderland manager insisted his captain had committed no offence.
"The first yellow card, nobody in the ground knows what it's for," he said. "Cameron Jerome fell over and made a meal of it but apparently he hasn't booked him for that but for Catts kicking the ball away.
"I've seen it and he never kicked the ball away, he passed it to them.
"You could say Catts was naive on the second one but in my opinion the referee, I think it's his second or third game, he was out of his depth and if you're going to fast-track people through they have to be exceptional."
Cattermole was involved in another contentious incident when he caught Birmingham striker Garry O'Connor with an elbow, causing a nasty cut above the Scot's eye.
Blues boss Alex McLeish was unhappy with the challenge, saying: "The guy's jumped with elbows outstretched.
"In the UK we still don't really penalise that but in the Champions League they can send you off for that."
Bruce, however, did not see anything malicious in the challenge and described the collision as a hazard of the job.
He said: "It's impossible to jump without your arms out. It's why I've got a face like I've got.
"If you're going to play this game then sometimes, when you jump for the ball, you're going to get caught."
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