Swansea, stung by recent reports of dressing room unrest, got just the tonic they needed ahead of upcoming fixtures against Manchester City, Liverpool and Chelsea 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."
Wigan boss Roberto Martinez, meanwhile, felt his side deserved to secure at least a point from the game, as their own winless run extended to six matches.
The former Swansea boss said: "I think we deserved something, it was the game we all expected in terms of the manner of both teams and the concepts we share.
"We saw a real fight for possession and who would control the game and I thought we did that better than Swansea in the first half. We created better chances and restricted Swansea really well.
"But in these games you need that sharpness in the final third and probably we missed that today [Saturday]."
Neither side wore the anti-racism campaign Kick It Out t-shirts prior to today's game. It was understood this was down to players looking to provide solidarity with their colleagues who did not want to wear them.
Laudrup admitted he had left the decision up to his players.
He said: "It's an individual thing because the issue is not only a football issue, it is social too. I don't think with a thing like that you can say you have to do it.
"It is better to ask each one of them (the players), I didn't want to have any influence on things, you have to ask each person as it goes much further than football."
Martinez said his side had opted not to wear them in support of Swansea.
He said: "We wanted to show a sign of respect to the home team, the home team for whatever reason decided not to wear them and I did not think it made sense for us to wear the t-shirts.
"We wanted to support the home team and I would expect the same thing if Swansea come to our ground and there is a gesture from the home team."