The Swans emerged comfortable winners thanks to a fine second-half performance following a turgid first half at the Liberty Stadium.
Teenage left-back Ben Davies got his first senior goal for the club before Jonathan de Guzman struck twice with a wonderful free-kick and a composed finish late on.
Michael Owen pulled a goal back late on for the Potters, the striker's first goal since October 2011, but it was a mere consolation.
The win tees Swansea up nicely for their second leg with Rafael Benitez's side, as they head into the game with a 2-0 advantage seeking a first appearance at a major Wembley final.
Laudrup said: "Winning is always the perfect preparation, winning and no injuries. I think it was a very good performance, we dominated from the first minute.
"In the first half Stoke had to defend very deep and the only bad point of the first half was we did not score from the two or three good chances we had, but apart from that I was pleased with everything.
"In the second half we continued, but I knew we needed something special to score the first goal, if we scored the first goal I was sure we would win.
"I obviously did not expect it to come from a run from the full-back, so it is great for Ben and he deserves everything for what he is doing and this will be a very special day for him.
"The second goal was a fantastic free-kick. I have seen Jonathan score a lot of these goals in training so it is nice to see it in a game and the third was the best of the afternoon."