Wayne Routledge's early goal and a last-minute Wilfried Bony header ensured Swansea picked up their first victory at Old Trafford by dumping out the 11-time winners in round three.
Sunday's result marked United's fifth home defeat of the season, but Laudrup claimed he was confident in his side's ability and did not rely on the Premier League champions' underwhelming form at their own ground.
"It's a great day for the players and the fans as well," the Dane said.
"It's a game we won, another game where we showed confidence, which we have done in recent games as well.
"It's obvious that Man United have got less points than they are used to, (but) for us it's much more important to go in and analyse what you can do and how you play.
"If you can do it against other big teams then you can do it here as well.
"(You need to) forget that they are wearing Manchester United shirts and just say 'we can do it', not (just) because four other teams won here."
Swansea were under plenty of United pressure for periods in the second half, but were helped by the dismissal of Fabio for a late challenge on Jose Canas 10 minutes from time.
Laudrup was delighted to see his side being clinical with their opportunities as they put David Moyes' men to the sword.
"To win against teams like this, you need to score your chances. You need, two or three times, a little bit of luck," he added.
"I don't think we needed that much luck today because I don't think Manchester United created as many chances as I expected them to."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.