Carlos Carvalhal's side needed a win, a Southampton loss and a 10-goal swing in order to preserve their place among the elite, having netted only twice in their last eight top-flight games.
Faced with that immense challenge, there was still room for sentiment as stalwart Angel Rangel started in his final outing for the hosts, with Leon Britton on the bench at the end of his long Swans career.
And Swansea – who look set to part company with Carvalhal – were soon celebrating when Andy King stroked home the opener, prompting chants of 'we want 10' from the Stoke end, with former Potters boss Mark Hughes at the Saints helm.
It was the first time the Welsh club had taken the lead in a match since the 4-1 victory over West Ham in March, which came towards the end of a run of 18 points from Carvalhal's opening 10 games in charge.
Having taken over in December with the side rock-bottom on only 13 points, the Portuguese looked to be leading a remarkable revival, but their demise was confirmed amid some signs of protest at the club's hierarchy.
Badou Ndiaye lobbed in the leveller and Peter Crouch scored what proved to be the winner before the break, converting the 1000th Premier League goal of the season, while Xherdan Shaqiri missed a penalty in a drab second half.
The two sides will next meet in the Championship, with Paul Lambert's men having had their fate sealed following last weekend's 2-1 loss at home to Crystal Palace.
To Swansea's credit, there was a vibrancy and attacking vigour to their first-half play and Jordan Ayew fired an early free-kick into the wall before an almighty scramble in the box, with Jack Butland standing firm.
But the England goalkeeper was wrong-footed when King collected Andre Ayew's flick-on and side-footed home to break the deadlock 14 minutes in.
An excellent opportunity to double that advantage followed within three minutes, but Wayne Routledge curled an effort wide from inside the box.
However, after Jordan Ayew saw a strike repelled by Butland, the visitors broke away to level through Ndiaye's confident finish following an excellent assist from Shaqiri.
And Lambert's side forged ahead before the interval, Crouch stooping to head in a cross from Lasse Sorenson, the youngest player to start a Premier League game for Stoke at 18 years, 204 days.
Jordan Ayew's fine free-kick had Butland scrambling across his goal, but the ball flashed just past the post to send Swansea in at the break wondering how they were behind.
Lukasz Fabianski prevented further damage with a save from Shaqiri's 54th-minute spot-kick after Mike van der Hoorn had handled in the box.
The remainder of the contest yielded further proof of Swansea's goalscoring troubles, with Tammy Abraham and Andre Ayew denied, while Britton's introduction was a welcome distraction from the pain of the home side's 21st Premier League loss of 2017-18.
Key Opta stats:
- Stoke secured their first league double over the Swans since 1992-93, when they were in the third tier.
- After saving just one of his first 15 penalties in the Premier League, Lukasz Fabianski has saved three of his last six in the competition.
- Crouch extended his Premier League record for most headed goals in the competition (53).
- Andy King has scored two goals from four shots on target for Swansea in 11 Premier League apps, this after scoring once for Leicester in 11 games this term.
- This is only the second time in Premier League history that two teams beginning with the letter ‘S’ have been relegated in the same season – also 1993-94 (Swindon and Sheffield United).
- Xherdan Shaqiri has had a hand in more Premier League goals this term than any other Stoke player (15 - 8 goals, 7 assists).