Substitute Manolo Gabbiadini brought Southampton close to safety as they saw off Swansea 1-0 at the Liberty Stadium, confirming West Brom's relegation from the Premier League.
The result means Swansea must hope they beat Stoke on Sunday and Huddersfield lose their last two games, while Southampton will be safe barring a defeat against Manchester City and a large goal swing towards Swansea.
Gabbiadini grabbed the winner with 17 minutes remaining after a scramble inside the penalty area, one of few clear-cut opportunities in a nervy encounter.
With so much at stake, the first half was a cagey affair, with half-chances falling to Sam Clucas and Charlie Austin inside the penalty area, but things picked up in the second.
Jordan Ayew forced a wonderful fingertip save from Alex McCarthy having cut inside from the left and unleashed a fine effort from 20 yards which was bound for the top corner.
Saints should have been ahead on the hour mark, but Austin's header in space eight yards out from Cedric Soares' cross was straight at Lukasz Fabianski.
But on 72 minutes they did break the deadlock as sub Gabbiadini, who had been on the pitch for just five minutes, drove home from close range after Alfie Mawson had blocked Shane Long's volley on the goal line.
Gabbiadini's substitution was a result of luck and a brave move by Mark Hughes, with the Italian striker replacing defender Jan Bednarek, who had concussion but did not want to be taken off.
Shane Long missed a chance to wrap it up late on, hitting into the side-netting when the square pass to the unmarked Gabbiadini seemed the better option, but after a nervy finish, Mark Hughes' side held on.
Man of the match - Alex McCarthy
After largely spectating in the first half, McCarthy was called into action early in the second, producing a world-class save from Jordan Ayew's dipping volley.
The 28-year-old, who replaced Fraser Forster as Saints' first-choice goalkeeper earlier this season, also commanded his area superbly, particularly late on as Swans peppered the box.
The managers
Carlos Carvalhal: "At the end of the game we had the feeling that we deserved a little more, a minimum of one point. But we played to win. Now we are in the position where we rely on others but we are not relegated yet.
Mark Hughes: "I have to say, to a man, they've worked their socks off, every single day, every single game. Some games we've had things go against us and have had to pick ourselves up, but I thought we were immense today, we weren't going to be denied."
Opta stats
- Swansea have failed to score in eight different Premier League home games this season, more than any other side.
- Seven of Southampton's last 11 away Premier League victories have been against sides starting the day in the relegation zone.
- Only Stoke (13) are on a longer winless run in the Premier League than Swansea (8 - D3 L5).
- Swansea have failed to score in eight of their 12 Premier League clashes with Southampton, scoring just five goals in total in those matches.
- Manolo Gabbiadini has netted two goals in his last four away league games for Southampton after failing to find the back of the net in the previous 12.
What's next?
Swansea host Stoke on Sunday in their final Premier League game, while Southampton welcome champions Manchester City to St Mary's.