The Swans required a late winner from Wilfried Bony to secure a 2-1 win at Old Trafford and book their place in the fourth round, but were good value for their success.
Wayne Routledge fired the Swans into an early lead, only to see Javier Hernandez restore parity within a matter of minutes.
It appeared as though the tie was heading for a draw and a replay at the Liberty Stadium until United substitute Fabio was shown a straight red card 10 minutes from time and the stage was set for Bony to produce a thrilling finale.
Laudrup was left beaming afterwards and claims the victory is just reward for everyone connected with Swansea.
The Dane said: "I think it's a great day for the players and for the fans as well.
"When you win at certain grounds, it is something very special for people who have only watched the Premier League for the last two and a half years.
"If you play Man United ten times away, maybe you win one or two, but the rest you draw or lose, which is normal.
"The reason why some stadiums like Old Trafford are special is because of the history of the club, not because of what happened in the last one or two seasons. To come here and win is always going to be something special.
"For us to beat them, you still have to deliver a very good performance."
Laudrup had seen his side go seven games without a win in all competitions heading into Sunday's testing trip to Manchester, but he claims confidence within the Swans camp has remained high throughout their wobble.
He added: "For me, it's another game where we showed confidence, which I think we have done in the last few games. We just needed a win, and today we got it in the last moments.
"Even if we have had six or seven games now without a win, my players play with a lot of confidence.
"Normally, when you have gone a few games without winning, you lack confidence, but I don't see that with my team.
"It was important that we tried to play our passing and movement game, and forget they are wearing the Man United shirts, and to just say 'we can do it'. Not because another four teams have won here, but because we think we can do it, like we did against Man City a few days ago.
"To win against teams like this, you need to score your chances. Maybe we got more chances today than I thought we would, and I don't think Man United created as many chances as I expected them to, but it was important to get the win.
"Bony's goal was a great finish, and apart from that I think we had two or three more chances. We only had 15 minutes in the first-half where we suffered a bit."