City have now taken 20 points from an eight-game unbeaten run and all Blackburn could manage in reply in Tuesday night's Premier League clash at St Andrew's was a late consolation from skipper Ryan Nelsen.
But McLeish is optimistic his team can build on their success during the first half of the campaign after being written off in August as certainties to return to the Championship.
The former Scotland boss said: "I'm proud of the players and I think the fans should be proud of them as well.
"They have shown a fantastic appetite and attitude since day one in the Premier League.
"They have also shown they can rise to the challenge when the expectation levels go up.
"With each game we have won, people are expecting more and mention things like Europe but we are keeping our feet on the ground.
"The Premier League is our main concern and we have given ourselves a very good chance to stay in the strongest league in the world.
"I never set any targets at the start of the season except trying to finish in the top 17. But we have got to be confident about our team and confident about building on this fantastic first half of the season."
McLeish was full of praise for Jerome who had only netted one goal before this double strike during the current campaign.
He said: "Cameron was fantastic. He scored goals against Blackburn here two years ago and before the game I thought he might score tonight. He was brilliant."
Blackburn boss Sam Allardyce felt his side handed victory to the home side through their own defensive errors.
He said: "We expected Birmingham to come out all guns blazing and put us under the cosh.
"We controlled the first 10 minutes but then give them a first corner and decide to give them a goal.
"With the form and confidence that they are in at the moment, it was always going to be difficult to come back from that.
"But to gift them another one really killed us. Apart from those errors it is a toss of the coin as to who was the better side but Birmingham were the ones who took advantage of their opportunities."