The Dane watched his side spurn several first-half opportunities at St Andrew's, Wilfried Bony and Alejandro Pozuelo both hitting the woodwork, before being made to pay in the second-half as their Sky Bet Championship hosts ran out 3-1 winners.
Laudrup was also left to reflect on missed chances in the club's past two matches, albeit a stunning 3-0 victory at Valencia and a 2-0 win at Crystal Palace, and he hopes his side now heed his warning ahead of Saturday's home clash against early Barclays Premier League pace-setters Arsenal.
"After the last two games where we played brilliantly, I said we won and had to be very pleased, but with the number of chances we created in those games we should have scored a little more. Here we paid the price for that," Laudrup said.
"I said after the Crystal Palace game, which we dominated for 90 minutes, it's only 1-0 (at half-time) and we should have scored more because even if you're dominating - one chance, one goal and you give your opponent belief. And that was exactly what happened against Birmingham.
"So let's hope this will be a lesson for us that we can use."
And Swansea defender Jordi Amat, one of 10 fresh faces called in by Laudrup for the third-round cup tie, has echoed his manager's thoughts after seeing the League Cup holders fall at the first hurdle in their defence of the title.
"It wasn't good. We had some chances to get that first goal but unfortunately we couldn't take one of them," he told the club's official website.
"If we had then I think it would have been different. The team is still in a good place right now despite the defeat, but it isn't always possible to win if you don't take your chances.
"We have a big game to look forward to on Saturday against Arsenal and we can learn from this defeat."