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 - who beat 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."
He added: "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."
Swansea will resume league business with a trip to Tottenham on Sunday, but they can head to north London elated at securing a first semi-final appearance in a major cup competition since 1964.
Laudrup added: "It's fantastic to reach a semi-final. I asked all the players before the game if any of them had played in a cup final, and no-one had. When you are three games from a final you get tense, and that is what we saw.
"Everyone can be proud we are in the semi-finals, but when you are there you want to go the next step and be in the final. Whoever we play next, it will be very special."
Middlesbrough must now pick themselves up for Saturday's home clash against Wolves, but they will only have one full day of preparation after a 320-mile journey home from south Wales on Thursday.
"Cup competitions are about getting through to the next round, and we didn't manage that," Boro boss Tony Mowbray said. "It was a tough game, fine margins. I could easily be sitting here having won 1-0, or having lost 1-0. Credit to them. They are in the semi-finals of the cup and good luck to them."
Source: PA