But the Tigers took the lead just past the hour when Liverpool loanee Lazar Markovic scored from close range after his header had bounced down on the goal line.
Hull made sure of the points nine minutes later when Sam Clucas found the top corner with superb 30-metre shot. They have now won four consecutive home league games. "I'm really happy. It was not easy. We played with 10 players, but our boys were fantastic and they deserved it," Hull manager Marco Silva told the BBC.
"I'm really happy but we still have four games to fight. Maybe in January no one would believe in our club to survive."
Spanish striker Fernando Llorente gave Swansea the lead with a trademark header from a corner in the 10th minute and the Welsh side were given a break when Marko Arnautovic missed a second-half penalty.
Swansea made sure of the win when Tom Carroll's shot found the top corner via a deflection off Joe Allen.
Time is running out for Boro who have not won in 16 league games and they endured a nightmare opening 20 minutes in which they conceded two goals and had midfielder Gaston Ramirez sent off for a second booking.
Josh King scored just 96 seconds into the match and Benik Afobe doubled the advantage for the Cherries.
Winger Marc Push made it 3-0 and Charlie Daniels completed the rout.
In Saturday's fourth clash, West Ham and Everton played out a dour goalless draw at the London Stadium, delivering a severe blow to the Toffees' hopes of an already unlikely top-six finish.