Wilfried Bony and Alejandro Pozuelo were both guilty of missing opportunities throughout the game, as lower-league Birmingham City ran out 3-1 victors at St Andrew's
The holders were knocked out of the competition in the third round thanks to goals from Dan Burn, Matt Green and Tom Adeyemi, with Wilfried Bony adding a late consolation.
Bony and Alejandro Pozuelo both hit the crossbar in the first half, with the latter missing two further opportunities, and Laudrup has rued his side's profligacy in front of goal.
"We could have been three or even four up at half-time, but we weren't and I said one day we will pay the price - I didn't know it would be today," Laudrup told reporters.
"They score the first time they have a chance and they are 1-0 up and then we have to score an equaliser and going forward like that saw us make mistakes and have misunderstandings at the back.
"The second and third goals were the result of our misunderstandings. We could and should have done better. We should have won this game after the first half. It should have been finished there.
"In the end we lost it, so I don't know what word to use but we can be a little sad about that.
"We won this last year and when you are champions you want to go a little bit further than this. It's the first game and we are out in a game we could and should have won."