The Championship outfit made life difficult for Laudrup’s men, with captain Grant Leadbitter testing stand-in goalkeeper Gerhard Tremmel on several occasions.
However, the home side got a slice of luck when Boro’s Seb Hines inadvertently headed into his own net in the 81st minute to help Swansea reach the last four of a major cup competition for the first time since 1964.
“It’s fantastic to reach a semi-final,” Laudrup said.
“I asked all the players before the game if any of them had played in a cup final, and no one had.
“Everyone can be proud that 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.”
He added: “It was a very tight game. 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.”