Defeat would have left Hull nine points behind 16th-placed Swansea but a second-half double from substitute Oumar Niasse helped them move above Middlesbrough into 18th place and within three points of the Swans and champions Leicester City.
With the top six, except Liverpool, all in FA Cup quarter-final action this weekend, the focus was on the survival scrap.
Bournemouth, one of the clubs too close to the bottom three for comfort, eased their relegation fears as Joshua King netted a hat-trick in a 3-2 home win over West Ham United.
Bournemouth missed two penalties and looked to have been denied their first win of 2017 when substitute Andre Ayew equalised for West Ham in the 83rd minute.
King, who drilled a ninth-minute spot kick wide, had the last laugh though when he struck at the death from a rebound after Darren Randolph had saved Jack Wilshere's effort.
"In those circumstances the main emotion is relief," said Bournemouth manager Eddie Howe, whose side are six points clear of the drop zone. "Missed penalties can happen but the most positive aspect is that we didn't go to pieces because of that."
Everton improved their chances of European football next season as they brushed aside West Bromwich Albion 3-0 with Romelu Lukaku rounding off a dominant display with their third.
Lukaku now has 19 league goals this season, level at the top of the scoring charts with Tottenham Hotspur's Harry Kane.
TOUGH GAME
Hull were bottom of the table when Portugal's Marco Silva took over in January but can now see light at the end of the tunnel after a victory that left them just one point behind 17th-placed Crystal Palace who were not in action.
Silva has presided over three league wins and with home games against West Ham, Watford, Middlesbrough and Sunderland on the horizon hopes are rising that he can keep Hull up.
"We got three important points, of course I am happy," he said. "It was a tough, tough game. We improved in the second half and after that we controlled the game."
There was a nervy finish after Alfie Mawson headed home for the visitors in added time but Hull held on for the points.
"I'm not happy with the last five minutes to give gifts to the opponent. The game only finishes when the referee gives the sign. But it is a fair result for us," added Silva.
Niasse, on loan from Everton, replaced Alfred N'Diaye after 63 minutes and he changed the game with two superb finishes.
Swansea's cause was not helped by a 33rd minute injury to Spanish striker Fernando Llorente, who had scored 10 goals in his last 15 league games.
Swansea, after a strong run to clamber out of the bottom three under new manager Paul Clement, are back in trouble.
"There's a lot of football to be played. I didn't think we were anywhere near safe and we are not anywhere near safe now," Clement said. "We have 10 games to go and have to bounce back next week against Bournemouth."
(Reporting by Martyn Herman; Editing by Ken Ferris)
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