A Ben Burgess brace has taken Hull City to within reach of automatic promotion.
He netted in the 60th and 87th minutes after the Swans had looked like salvaging a point.
"We deserved all three points and now I reckon another three should secure promotion," said Tigers boss Peter Taylor.
"Perhaps we should have scored a couple more in the first half but I knew it would be a difficult game.
"I have known Kenny Jackett for a long time and I think he is a tremendous managerial appointment for Swansea.
"He will make sure his players work hard throughout every match and the club will surely benefit from his coaching experience." The Tigers took the lead in the 10th minute when Burgess threaded a neat pass through to Danny Allsopp, who finished with a crisp shot into the corner.
Swansea levelled on the stroke of half-time when an Andy Robinson shot beat the diving Boaz Myhill in the Hull goal.
Jackett's side started the second half on top, but a howler by veteran keeper Roger Freestone on the hour gifted the visitors the lead when he let a Burgess chip slip through his fingers.
Swansea thought they had rescued a deserved point when Lee Trundle equalised in the 82nd minute, but Burgess popped up with three minutes remaining to grab all three points for Hull.
It was a disappointing home debut for new boss Jackett.
"After we got back into the game twice it was a pity we lost out, but there were plenty of positives.
"We had a real go at them and caused them problems and I am delighted with the contribution from all our young players.
"I saw some pride and passion out there, which looks good for future prospects.
"I will assess the squad fully before deciding what to do in readiness for next season, but first and foremost I want results and I want the club to be successful."