The Swans reached only a third semi-final in top-flight cup competitions in their 100-year history thanks to an 81st-minute own goal from Boro's Seb Hines that secured a 1-0 verdict at Liberty Stadium.
It put Swansea into the last four alongside Bradford - memorable conquerors of Arsenal - and Aston Villa, while Chelsea meet Leeds in the remaining quarter-final clash next week.
"It was a very tight game," Laudrup said. "It wasn't a brilliant game to watch for the neutral, but it was a tight cup game.
"Everybody had the right attitude, everybody wanted to win, but I think they wanted it too much (in the first half). When you want something too badly, you can't do exactly what you usually do.
"Middlesbrough put pressure on us, but in the second half we came out and played much, much better. We were more relaxed and took control of the game.
"It wasn't a game with a lot of great chances. At Crawley earlier in the competition, we were 2-1 down and won in the last minute, and the Liverpool game was tight. It happens a lot in cup games."