The win moves Mark Hughes' side into 16th place in the Premier League – ahead of Huddersfield Town on goal difference – and they know they will be safe if they avoid defeat against Manchester City on the final day of the season this Sunday.
It is looking increasingly bleak for the 18th-placed Swans, who need to beat Stoke City at the weekend and hope that Huddersfield fail to pick up any points against Chelsea and Arsenal in their remaining two games or that Saints lose heavily to Pep Guardiola's Premier League champions.
Substitute Gabbiadini struck in the 72nd minute after pouncing on a loose ball in the box – a goal that also confirms West Brom's relegation to the Championship.
Chances were at a premium in a hard-fought affair, with the clearest regularly falling to Charlie Austin, who found Lukasz Fabianksi equal to everything he could throw at him.
Swansea pushed forward in numbers in the final ten minutes but they were unable to find a leveller that would have made their task on Sunday much less daunting.
In an opening played at breakneck speed, it was Swansea who carved out the first half-chance as Andy King drilled a low effort straight into the hands of Alex McCarthy after eight minutes.
Jordan Ayew then headed over from an inviting position midway through the opening 45 minutes before Austin came close to breaking the deadlock.
Pierre-Emile Hojbjerg found the striker with an incisive throughball, but his angled drive towards the far corner was too close to Fabianski, who got down well.
The influential King was involved again shortly before the interval when his clipped cross was met by Sam Clucas, whose header flashed past McCarthy's far post.
Austin continued to look the most likely to find a breakthrough for the visitors and he again threatened with an acrobatic effort from Cedric Soares' cross, which was well held by Fabianski.
Swansea's intentions after the restart were clear, with Jordan Ayew denied a spectacular opener when McCarthy tipped his whipped long-range effort over the crossbar.
Fabianksi was once again equal to Austin's latest effort on target - this time a dipping strike from just outside the penalty area.
Their fascinating duel showed no signs of abating as the Polish goalkeeper kept out Austin's header on the hour, while substitute Tammy Abraham drilled straight at McCarthy after a determined run.
Perhaps unsurprisingly, Austin and Fabianski were involved in Gabbiadini's winner. The Swans stopper repelled Austin's strike from a corner, but the ball fell kindly to the Italian who reacted quicker than a host of defenders to stab home.
Carlos Carvalhal's side huffed and puffed in the closing stages, but the Saints backline held firm to hold on for a win that all-but secures another season of Premier League football.
Key Opta Facts:
- 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.
- Gabbiadini's goal was just his fifth in his last 41 games for Southampton in all competitions, one fewer than he managed in his first four games for Saints.