Swansea found themselves 1-0 down at half-time after Blues striker Lee Novak headed the Championship side into the lead after a fine cross from Paul Caddis.
With the deficit in mind, Laudrup sacrificed Pablo Hernandez at the break and introduced Bony, which proved to be an inspired move.
The Ivory Coast international came on and scored two goals in as many minutes to cancel out Birmingham's lead as the Swans edged their way into fifth round.
After a difficult first-half, Laudrup was pleased with his side's response.
"A win is always good for confidence, especially in the situation we are in," the Swansea boss said. "The reaction was pleasing as well.
"The first half was very difficult, we lacked a bit of confidence. We really wanted to win, but sometimes you play a bit too nervously.
"We spoke about it at half time that if you are that tense, it is difficult to play football."
Laudrup's team talk and tactical changes during the break proved to be the difference at St Andrew's.
With Swansea hosting Fulham at the Liberty Stadium on Tuesday night, the Danish manager is keen to see his side build on their second-half performance as they look to end a nine-game winless streak in the top flight.
"We saw a much more relaxed team in the second-half, created a lot of chances and scored two goals. We started to play with confidence again.
"It's important for the players ahead of some crucial games against directing opponents, starting with Fulham on Tuesday.
"This result doesn't change what we have to do on Tuesday, but it gives us something because it is a win. A win always gives confidence."
Laudrup was quick to pile the praise on Bony following his brace, claiming he was the perfect counter to Birmingham's style of play.
"With the way that they played, we had to start Bony from the second-half - and it paid off," he added.
"He has had to adapt to a different culture on and off the pitch. It is always difficult when you are a foreigner.
"But now that he has adapted, he is playing with a lot more confidence."
Bony brought Swansea level when he smashed home Roland Lamah's cross before sealing the tie two minutes later by comfortably finishing a slick piece of play.
"The first goal was a great goal, and the second, he wouldn't have scored if he didn't have confidence," he said.
"I am happy for him and for the team. Hopefully we can build on this result today."
Birmingham manager Lee Clark also lauded the efforts of the Swansea striker, but was left to rue not scoring a second when they had the chance.
"He's in very, very good form at the highest level of the game," said Clark. "We knew we had that to contend with if we went in front.
"I'm a bit disappointed because, when we had the ball, we were very good, but we killed ourselves with the second goal.
"It's something we spoke about and it could have been prevented.
"We did extremely well with the ball, but when you play against a Premier League side and switch off, they will hurt you."