Unfancied Swansea had won just once in their last ten games before yesterday and, despite United's struggles under David Moyes, they were written off by most ahead of their third-round trip to Old Trafford.
And though they were without a host of injured stars — and chose to leave a number of regulars on the bench — much-changed Swansea chalked up one of their great victories.
Swansea led early on through Wayne Routledge, but his classy finish was quickly cancelled out by Javier Hernandez.
Yet after United defender Fabio was sent off late on, Swansea won it in the 90th minute thanks to Wilfried Bony's header.
It was Swansea's first ever win at the Theatre of Dreams, and their first over United since 1982.
And Laudrup said: "First of all this is for the players and the fans. When you win at certain grounds it is special for them.
"Last year we had the Emirates and Stamford Bridge, and now we have Old Trafford.
"It must be something very special for people who have only been watching Premier League football for two-and-a-half years.
"It's a great day for them and for my players, who played with confidence again.
"For me they have done that in the last few games as well. We just needed a win and here we got it in the last minute. That was the difference.
"The most important thing for me is seeing my team play with confidence. The day I will be worried is when I see four or five players looking shaky on the pitch, and I haven't seen that yet."
Swansea became the latest in a line of clubs to inflict home defeat on a United side who are struggling to adapt to life after Sir Alex Ferguson.
But Laudrup insisted: "In my world, they are still a big team and for us to beat them we still have to produce a very big performance.
"The reason some stadiums — Old Trafford, the Bernabeu or the Camp Nou — are very special is because of the history, not what has happened in one or two seasons.
"Winning at a ground like this is not something that will happen every day — it could be 15 years before Swansea do it again — so it will always be special.
"If we play at Old Trafford ten times, we might win one or two and draw one or two and the rest will be home wins.
"What we had to do was try to play our passing football, like we have against the other big clubs.
"You have to try to forget their players are wearing Manchester United shirts, not because of what anyone else has done here but because we believe we can play our game at a ground like this."
Swansea's reward for yesterday's memorable win is a rather less glamorous fourth-round trip to either Birmingham, Bristol Rovers or Crawley.
Rovers and Crawley are still to play a second-round replay — it was scheduled for this weekend but was called off due to the weather — with the winners of that game to battle Birmingham for the right to host Swansea.
"It's a little curious that we could play Crawley, who gave us one of our toughest games on the way to Wembley last season, or Birmingham, who knocked us out of the Capital One Cup this season," Laudrup said.
"That just shows you what can happen in the cups. We can win at Manchester United but lose there."
Swansea are today waiting for news on the extent of Roland Lamah's groin problem after the winger became the latest player added to their injured list.