The loss marked a six successive defeat for the struggling Robins to leave them stuck in the bottom three of the Championship.
McInnes made no excuses for his side and emphasised the importance of their clash against Charlton this weekend.
The boss said: "This is now six defeats in a row.
"It is embarrassing. I am embarrassed. The players should be embarrassed and it is now a crucial game against Charlton on Sunday."
City had two players - Cole Skuse and Steve McManus - badly injured in the St Andrew's clash, and the visitors had to play out the final 10 minutes with only 10 players.
Skuse suffered a fractured cheekbone and damaged his eye socket and was taken to hospital.
He is expected to be out of action for two months, while McManus could be ruled out for the rest of the season with a cruciate ligament injury.
McInnes added: "The game ran away with us with the injuries at the end. We cannot hide behind the fact that the better team won on the night."