Swans won thanks to two goals in three second-half minutes from Pablo Hernandez - the Spaniard's first for the club - and Michu.
Emmerson Boyce pulled one back for Wigan and Arouna Kone had a header ruled out for a tight offside decision as Swansea retreated ever deeper defensively, but they held on for a vital three points.
Laudrup said: "You can say there is a mixture of joy and relief, we knew today was very important for us and I think if you see the game the players played with a lot of confidence and patience.
"But you know when you really want something, in this case a win, you cannot think you can do it in 10 minutes. At this level it takes 90 or 94 minutes and that is what happened.
"We knew both teams would want the ball as they are both good in possession and I think in the first half we had a little more possession and felt comfortable.
"But from the first minute of the second half we dominated completely, we did not create loads of chances but we got closer and closer and suddenly it was there.
"There was one goal and then we got the second and at that point everyone was thinking when would the third come to seal the win.
"But it sometimes happens in football; they got a goal back and the game changed completely and we had to suffer for the last 20 minutes.
"They had a goal disallowed which was very, very tight but to come back after some games without winning it would be too easy to win without suffering."