The Swans had not won on home soil in eight Barclays Premier League games and were frustrated by the Black Cats during a goalless first half.
But Phil Bardsley's own goal, a wonderful Jonathan de Guzman strike and Wilfried Bony's penalty - Swansea's first in 56 league games - completely altered the course of the match in just seven minutes.
Chico Flores rubbed further salt in the wound by heading in a De Guzman corner - via unlucky Sunderland striker Steven Fletcher - with 10 minutes to go.
Swansea had only been above the relegation zone on goal difference heading into the weekend and their tension at the lack of a home league win since March was clear to see during an error-strewn first half.
But Laudrup was delighted by how his side had turned it around after the break.
Laudrup said: "In the Premier League it has been a while since we won at home. We have won all three of our home games in the Europa League, but this is our first of the league season.
"We can always talk about who we have played against, and we have played three of the top teams here this season.
"But when people start to talk it is good to turn it down immediately, and the only way to do that is by winning."
The only negative for Swansea was an ankle injury suffered by left-back Ben Davies, who was withdrawn early in the second half and left the Liberty Stadium wearing a protective boot.
Laudrup confirmed the Wales international will undergo an X-ray to determine the severity of the damage.
He said: "It's another ankle injury, we have already had a few.
"It is a twisted ankle, he fell in the first half so we agreed to let him start the second half but after two or three minutes you could see it wasn't good.
"He will probably go for an X-ray, but I am not a doctor and we have to wait for the X-ray. But I think he may be out for the next two games (against Kuban Krasnodar and West Ham) at least."