The Dane saw his side pick up a 3-1 win on Merseyside on Wednesday night to dump out the League Cup holders and book a quarter-final home date with Middlesbrough.
Goals from Chico, Nathan Dyer and Jonathan de Guzman did the damage, with Luiz Suarez scoring a consolation for former Swansea boss Brendan Rodgers' Liverpool.
And after spending his childhood supporting the all-conquering Liverpool of the 1970s, 48-year-old Laudrup said: "It is most important to go into the quarter-finals.
"But beating a great team here in the special arena that is Anfield is very special.
"It is for me, I am a foreigner. But imagine for the players, for the club and the fans. It has been really great."
Laudrup now hopes Swansea can take self-belief from defeating Liverpool as the Welsh club prepare to host Chelsea in the league on Saturday.
The Liberty Stadium club currently sit 11th in the top flight but they have won only one of their last seven matches, although they were unfortunate to lose at Manchester City last weekend.
And Laudrup said when asked about the impact of winning at Anfield: "I do not know what it means for the season. When you get down the road, you have to analyse whatever we have done.
"But, right now, it gives fantastic confidence and something special to everybody - club, players and fans."
He added: "We also know the league is another thing. We have to get points to build on. We have to get points to play one more year.
"But I think if we go on like we have played in the last games, I think we will have a comfortable place in the competition (the Premier League)."