Kelvin Davis' error allowed Mame Diouf to equalise Morgan Schneiderlin's first league goal since September and Adam delivered the knockout blow for Stoke six minutes from time.
Two weeks ago Adam scored a stunning long-range strike from his own half against Chelsea
On Saturday he fired in from much closer range to seal this 2-1 win.
That is not to say the winning goal lacked quality
The Scot, who came off the bench at half-time, swivelled cleverly away from his marker and fired a low strike under Davis after the ball had bounced around the box.
The win, Stoke's first in five matches, means they edge to within four points of their 50-point tally for the season
They have no goal left to aim for in their final five matches, but manager Mark Hughes can take great encouragement from the heart his men showed.
For Southampton, their dreams of qualifying for the Champions League appear remote
They remain in sixth, five points behind fourth-placed Manchester City, who play against West Ham on Sunday, and one point behind Liverpool.
And Ronald Koeman's problems worsened at the death when key defender Toby Alderweireld was carried off on a stretcher with a leg in a brace after suffering what looked like serious injury after falling awkwardly.
Stoke looked very much like a team that had failed to win in six weeks during the first half.
Apart from an early opportunity for Steven Nzonzi, which he blazed high and wide, they could not fathom any decent openings.
Southampton were playing far more confidently, turning defence into attack thanks to some precise passing and impressive running from Sadio Mane.
The pacy Saints midfielder thought he should have had a penalty in the 11th minute when he was bundled over by Stephen Ireland, but referee Mark Clattenburg waved play on
The home fans thought Mane had dived and yelled "cheat, cheat, cheat" at the Senegalese.
Jose Fonte saw yellow for a late tackle on Diouf during a rare attack from the hosts.
Saints were soon back on the attack
Mane turned and slid in Graziano Pelle, and he fired a fierce shot into the side-netting.
The visiting fans were mocked for thinking the ball had gone in
Four minutes later they were cheering a real goal.
Fonte met Steven Davis' corner and glanced it past Asmir Begovic
The defender was just about to start celebrating his first goal of the season, but Schneiderlin nicked it off him by smashing the ball in from less than a yard.
The Southampton fans started singing about Europe, but they had to finish the job here first.
Nathaniel Clyne wastefully ploughed a shot into the stands when Pelle was free at the back post and the right-back then went into the book for clattering into Erik Pieters.
Nzonzi was handed an opportunity to equalise on a plate just before the break, but he shinned wide from two yards wide following Geoff Cameron's flick-on.
Stoke looked a completely different team in the second half
Adam replaced Ireland at half-time and within two minutes the hosts had equalised.
Davis, who had barely had a thing to do all match, should take a large portion of the blame for the goal
The Saints stopper was stuck in two minds when deciding whether to catch Nzonzi's shot
He could have easily gathered the ball, but he decided against it
The ball struck the angle and fell straight to Mane, who lashed in from close range.
Davis' opposite number then almost gifted Saints the lead
Begovic came racing out of his box, but Mane beat him to the ball and squared to Dusan Tadic, but his weak shot was cleared off the line by Glenn Whelan.
The match was a real end to end affair with both teams going for the winner.
And Adam made sure of the victory for the hosts with six minutes to go
After a sustained period of attack from Stoke, Adam nicked the ball off his marker and swept home after Jon Walters' shot was blocked by Fonte.
Substitute Peter Crouch missed a great chance to score a third, but the game had already been won.
Then things went from bad to worse for Saints when Alderweireld was carried off with what appeared to be a serious injury
Source : PA