Cameron Jerome grabbed the crucial opener in first-half stoppage time, capitalising on some hesitancy in Wolves' six-yard box to bundle the ball over the line with Wayne Hennessey managing to get just a single hand on it.
Garry O'Connor then made the game safe in the 69th minute - again amid more confusion in the Wolves defence.
Christophe Berra's attempted header back to Hennessey fell short and the striker pounced to round the Wolves goalkeeper and slot home.
"The players are euphoric in there, it was a great performance," said McLeish.
"We had everything tonight - with the drama in the game, great goals and a superb work rate and organisation from my team.
"They were absolutely magnificent and I have to compliment the full house at St Andrew's, it was a joy to behold and they played their part as well."
He added: "It's three huge points, we close the gap on Wolves and we've got to make an attempt to win the title."
Mick McCarthy felt his side were "the masters in their own downfall".
But he was also left to question the validity of Jerome's opener.
"With the first goal I wonder whether there's a foul on Wayne Hennessey?" said the Wolves boss.
"I actually didn't think it was a free-kick he penalised Marlon Harewood for on the halfway line (in the build up to the goal).
"The second goal was a schoolboy error and we've been the masters of our own downfall.
"We've lost points, I don't there was any quality in the game. The first goal wasn't really and neither was the second."
Despite the importance of the result, the game was somewhat overshadowed by Lee Carsley's foul on Chris Iwelumo in the 36th minute - which resulted in the Birmingham captain seeing red.
The challenge, which brought back memories of Martin Taylor's foul on Eduardo, was dangerously high on the Wolves striker and resulted in him being stretchered off.
"Chris has got a sore knee but we'll know more tomorrow. He's got it in ice but he'll probably go for a scan," confirmed McCarthy.
"At the time I couldn't really see it, but I've just watched it again and let me clear something up.
"Lee wouldn't go in to hurt anybody and that's for sure, he's a competitor and I think he's just mistimed it, he's caught him and it's not a very nice tackle.
"I don't think there's any doubt it was the turning point, the game was going nowhere. I thought we were comfortable except for a couple of mistakes."
Blues boss McLeish echoed his opposite number's thoughts on Carsley and the challenge.
"I thought he wasn't the favourite to win the ball. Lee's a wholehearted player, committed, he's not a dirty player by any means and all the professionals know that," said the Blues boss.
"He was upset at half-time. He knows that the player is injured and he cares, he is a guy who cares not only about the cause for Birmingham but he respects his fellow pro's."