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 period 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."
It is a far different story for a Birmingham side who have now gained plenty of momentum with impressive back-to-back wins following an indifferent start to the season.
Jesse Lingard, unavailable against Swansea due to the terms of his loan from Manchester United, provided a breath of fresh air as he became the first player in over 60 years to mark his Blues debut with a hat-trick as he netted all four goals in last weekend's 4-1 triumph over Sheffield Wednesday.
And that has been quickly followed up by registering the upset of the third round to provide Birmingham fans with some welcome cheer, having endured a fall from grace - both on and off-the-pitch - following their own League Cup success just two years ago.
Centre-back Dan Burn and midfielder Tom Adeyemi netted their maiden goals for Lee Clark's side to help down Swansea - Matt Green chipping in with the other - and the pair both feel Birmingham are now brimming with belief.
Burn said: "We've played really well all season, taking away the Burnley game (a 3-0 defeat), but now we're starting to get results you can see the confidence is starting to build around the team and lads are more confident on the ball.
"Different players are starting to score, Matt Green has got a few goals now and he looked an absolutely great player here. So it's nice we're playing with confidence and that always helps.
"Jesse came in and did really well on Saturday and showed the lads that we could take our chances and we're going to get results, and we've done it again."
Adeyemi added: "I think prior to these two results we were still confident because our level of performances have been reasonably good.
"We'll take a lot of confidence out of the fact that we've scored seven goals in two games because that was one area in which we were struggling a little bit.
"Jesse's probably been the catalyst for that before then moving on to this win."