The Swans' poor recent run looked set to continue when Lee Novak's glancing header gave Birmingham a deserved lead in this fourth-round tie.
But the introduction of Bony at the interval changed the game, with the Ivorian scoring twice in three minutes to take his tally to 16 for the season.
Victory means they can approach Tuesday's key Premier League fixture against Fulham with a positive result behind them at the end of a week dominated by the training ground row between Chico Flores and Garry Monk, and reports of divisions within Laudrup's squad.
And the Dane admitted Bony, who has now scored 16 goals this season after arriving in a £12million move from Vitesse Arnhem, had got his side out of jail.
The Swans boss said: "I thought the way they played in the first half meant we had to bring on Bony and he scored two great goals to turn the game.
"We talk about the need to adapt on the pitch and it is not easy as a foreign player. I have been a foreigner in most of the counties I played in and I understand that.
"But when you have a big price tag it is about patience and I am happy for him and the team.
"He has a cool head, you can see how he finished his goals, he is playing with confidence. The first goal is a great finish, the second goal is one you do not score without confidence and that is very good to see.
"Hopefully we can build on this result in the game on Tuesday against Fulham."
Laudrup also felt Swansea's recent run had affected their first-half performance.
He said: "When you are on a difficult run you can be tense and you could see a lack of confidence on the ball.
"We were too tight and tense and that makes it difficult to move the ball.
"At half-time we said we needed to relax and we were better in the second half and scored two good goals."
Birmingham boss Lee Clark also praised Bony's contribution but felt his side had missed a big chance to cause an upset and reach the fifth round.
He said: "Bony is in very good form at the highest level, I watched him over the last few games and he looked very, very in-form.
"You know you have that to contend with if you get in front. It is a huge game for them on Tuesday and we are disappointed.
"His first goal was very good but the second goal we could have done more to prevent.
"You always need that second goal against a Premier League team especially with their fire power, we had chances, but we weren't able to take them."
Clark also said the decision to omit Nikola Zigic from his matchday squad had been for "football reasons".