Birmingham started on the wrong foot and were on the end of a third-minute opening goal from Luke Beckett, who grabbed his third goal in this season's competition with a simple tapped finish.
Garry O'Connor replied with an equally straightforward strike following good work from Fabrice Muamba but Town refused to settle for a replay and were rewarded when Chris Brandon converted Beckett's deep cross.
For McLeish, in his first management role south of the border, it was a grim introduction to the competition.
The former Rangers and Scotland boss admits he was desperate not to be on the receiving end of a giant-killing and has called for his players to take a firm look at their personal contributions to defeat.
"The Premier League is our priority and it remains our priority but we wanted to win this match, no question," said McLeish.
"I've been involved in cup upsets as a player and a manager. I've been a giant killer but I've seen the other side and I didn't want to be a victim today.
"The players will be embarrassed by it because they'll get bad headlines and so will I. That is the position we are in."
Reflecting on what had been said to his off-colour side behind closed doors, he added: "The dressing room chat remains private and that is where it must stay.
"But every one of our players can look in the mirror and say 'we've let ourselves down and let the fans down'.
"We had three or four internationals out there and we should be better than that."