Summer signing Michu broke the deadlock after just eight minutes and doubled Swansea's tally in the second half, netting an exquisite curling left-foot strike.
The old guard wrapped up the 5-0 victory as Nathan Dyer grabbed a brace and Scott Sinclair struck to get Laudrup's life at the Swansea helm off to the best possible start.
"That was impressive," the Dane said. "Winning the first game in my debut - and also the team's first in their second season in the Premier - away and 5-0. I don't think anyone could've dreamed about that.
"Of course, the most important thing is the team and I am very happy for the players.
"Last season I think it took until January to win the first away game and it is very important that you know you can win an away game as it gives you a lot of confidence for the rest of the season.
"If you know or think you will always lose or maybe only draw, you have to win all your home games.
"It has given us a lot, a lot of confidence, which we can bring to Swansea for the next two games to try win at home."