Local boy James Thomas struck a hat-trick to help Swansea retain their league status for next season.
He netted penalties in the sixth and 44th minutes and then sealed victory with a superb chip over the Hull keeper 12 minutes into the second half.
"It was the most important game of my career and to score a hat-trick in a match like this is unbelievable," said Thomas.
The Swans started well and should have calmed nerves when Thomas dispatched the first spot-kick after Leon Britton was brought down in the area.
But defender Lee Jenkins was caught in possession by Stuart Elliott three minutes later and he easily beat Neil Cutler from close range to level the scores.
Worse was to follow for the home side when fellow defender Michael Howard was dispossessed by Martin Reeves who went on to fire home with just 25 minutes gone to put Swansea 2-1 down.
As a draw for Exeter would keep the West Country side up at Swansea's expense if the scores remained the same at the final whistle, it was imperative that they got back on level terms as quickly as possible.
They were given a lifeline one minute before the interval when a Hull defender handled in the box and Thomas stepped up to calmly hit the net for a second time from the penalty spot.
"I just went for the corner and the second one was really crucial as it meant we went in all-square at the interval." The Swans were now on a roll and Lenny Johnrose restored the lead three minutes after the restart.
Thomas completed a memorable hat-trick in the 57th minute when he chipped Northern Ireland international keeper Alan Fettis from fully 30 yards to make the game safe for the Swans.
"I don't know how I managed it," said Thomas, who was born just a few miles from the Vetch Field and who used to support the team on the terraces as a young lad.
"I was feeling confident after scoring twice and just decided to try a chip." Brian Flynn has had all his hard work rewarded by this victory.
"I thought we would do it, but I knew there would be a lot of twists and turns even from a long way out.
"For a local boy who supported the club as a youngster to score a hat-trick in such a vital game is incredible. I knew it was a challenge when I came here, and that is exactly what it has been."