Steven Fletcher scored two first-half goals on his debut at the Liberty Stadium as the Black Cats tested the home side's resolve.
Laudrup said that the point his team earned meant as much to him as the opening two wins of the season.
"I said to the players after the game this is the game that has given me the most because everything was against us from the beginning," Laudrup toldBBC Sport.
"For me, this point is much more than a point for me. It shows me that the team has a lot of character, because in the first two games everything has been for us. So this was very important for me to see what happened today.
"[The red for Chico] was very significant. You get the 2-2 and then you have 20 minutes left and then you get a red card. I know Chico, I know he goes for the ball. It's true, it is a high leg. I thought for sure it would be a yellow card. I just hope the referee looks through the DVD a couple of times before he makes his report."
Swansea travel to Villa Park for their next Premier League fixture.