The Jack Army needed a late goal from Ki Sung-Yeung after several failed attempts following the introduction of Andre Ayew and Tammy Abraham.
The Korean latched on to a pass from Kyle Naughton and sent his effort past goalkeeper Nick Pope for the match-winner.
“I am very proud of my players,” Carvalhal told the club's website.
“We played against a difficult team to beat, with the third best defence in the league, and we knew it would not be easy to score against them.
“But we felt in the second half that my players wanted to win a lot and we tried to help them by putting more attacking players [like Andre Ayew and Tammy Abraham] on.
"We took some risks, but they were controlled risks because we have trained this way. There is a famous coach in Portugal, Quinito, who says ‘we put all meat on the barbecue to try to win the game," he said.
“That is what we did today. We had Kyle Naughton in the right corridor and Sam Clucas on the left, then three forwards.
“We thought we could win and in the end I think we deserved it.
“We could have been happy with a point and stayed as we were, but we put players on to try to win and we did it,” he added.
Swansea have moved up to 15th position with 27 points, two points above safety. They will face Sheffield Wednesday in the FA Cup by midweek.