Jordi Gomez celebrates his extra-time penalty winner for Swansea
Hull were unable to grasp the lifeline offered when Swansea missed a late penalty in normal time and bowed out of the Carling Cup after an extra period.
The Premier League visitors took the lead when Caleb Forlan dummied a Nick Barmby pass for Dean Windass to score.
But Gorka Pintado glanced a header past Matt Duke for a 63rd-minute equaliser.
Pintado should have won it with a spot-kick in injury time, Duke saving but unable to stop Jordi Gomez's extra-time penalty after Michael Turner's foul.
Championship new boys Swansea were behind after 11 minute when Barmby fed the ball through to Folan, who stepped smartly over it for 39-year-old Windass to finish coolly with his left foot.
Swansea were unlucky to see penalty claims waved away when Hull skipper Wayne Brown appeared to push Guillem Bauza.
Referee Mike Russell gave the decision against the Spaniard, ruling he fouled the Tigers captain on his way into the visitors' box.
Bauza wasted a glorious chance to get the Swans level on 56 minutes, firing over from 12 yards when substitute Pintado set him up.
Pintado then forced a fine save from Duke, who parried his scissor-kick effort before having another header cleared off the line.
But on 63 minutes "El Toro" finally got his goal, heading home from Gomez's floated ball into the box.
It was Swans keeper Dorus de Vries' turn to pull off a great save moments later from Ryan France, then back up the other end as Pintado crashed a shot against the Hull post.
In injury time substitute Febian Brandy won a penalty after being fouled by Brown, but Duke read Pintado's spot-kick.
Yet Swansea got a second chance to settle it from the spot 14 minutes into extra time.
Gomez sent a ball through to Shaun MacDonald, who was brought down in the box by Michael Turner.
Up stepped Gomez and he made no mistake from the spot for his first goal of the season.
Swansea manager Roberto Martinez: "In the first half Hull City showed why they are a Premier League side.
"Probably, we gave too much respect and were a bit cagey and we paid the price.
"The reaction we had in the second half was very pleasing, we controlled the game against the Premier League side which shows you everything.
"Gorka Pintado could have had a hat-trick and I thought it was a fantastic night of football."
Hull manager Phil Brown: "Fair play to Swansea, they played some magnificent football.
"We were looking in the first half like we were an organized team. In the second half, that organization went out of the window.
"We lost our shape and discipline which was very uncharacteristic of us.
"Swansea got hold of the ball and the game and they went on to win deservedly."
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