First-half goals from Fernando Llorente and Kyle Naughton handed Swansea their third win in four Premier League games, leaving them four points clear of 18th-placed Hull – who travel to Crystal Palace on Nissan Super Sunday.
The result means Hull will go down if they lose to Palace, while a draw is unlikely to be good enough given their inferior goal difference.
Meanwhile, the Swansea players – who had paid for their fans to attend the match in the north east – ensured their 3,000 supporters left the Stadium of Light elated as they edged ever closer to safety.
It was just the one change for Sunderland as Seb Larsson came in for the injured George Honeyman, while Paul Clement kept faith with the same Swansea XI which beat Everton last weekend.
The visitors were on top from the off and were rewarded for their bright start when Llorente headed in his 14th league goal of the season.
It came from another Gylfi Sigurdsson assist – his 13th this campaign – with the Icelandic’s ball in met by Llorente ahead of Sunderland goalkeeper Jordan Pickford, who came rushing out to try and punch clear.
Despite a lacklustre first half from the hosts, they did have a couple of opportunities to equalise – firstly through Billy Jones, but he could only head wide from 10 yards out.
Jermain Defoe was then denied from a narrow angle, as Lukasz Fabianski spread himself to deny the Sunderland forward following Jones’ knockdown.
The Swans upped the ante towards the end of the half however, and were rewarded when Naughton flashed an effort into the top corner from a tight angle in the third minute of added time.
That was Naughton’s first ever Premier League goal for Swansea in what was his 141st appearance, and it left the hosts with a mountain to climb after the break.
Sunderland grew into the game in the second half, and they should have had a penalty when Sigurdsson stuck out an arm to block Seb Larsson’s free-kick – but referee Andre Marriner waved away the Swede’s furious protests.
With 20 minutes to go, Sunderland’s Jones missed a gilt-edged chance as he could only fire agonisingly wide after Fabianski spilled Khazri’s free-kick.
Swansea failed to register a shot in the second half, but they withstood everything Sunderland threw at them to claim their first away win since January.
Sunderland meanwhile went out with a whimper in their final Premier League match at the Stadium of Light for at least a year, while striker Jermain Defoe could be one of a handful of players to have played his final home game for the club.