Frenchman Neal Maupay gave Brighton the halftime lead but Danny Ings pulled one back in the 66th minute for Southampton, who kept up a furious second-half onslaught but were unable to convert it into a come-from-behind win.
Brighton moved up to 37 points, six above the relegation zone with two games remaining and a vastly superior goal difference but still need another point to be certain of survival.
Southampton remained in 12th place on 46 points, keeping up the unique distinction of being the only club with a better away record than home in this season's Premier League. They have earned 28 points away and 18 at St Mary's.
But while the visitors deserved their half-time lead, they spent most of the second half on the back foot as Southampton came out with a lot more endeavour.
Vokins then hit a driving shot from well outside the penalty area that was just inches away from a goal on his debut.
The pressure continued to mount as Ings saw his 56th minute shot hit the post and bounce back for Adams, whose effort on the rebound was cleared off the line by Lewis Dunk.
Adams was then just wide with a sliding effort on the end of a square pass from Ings and Pierre-Emile Hojbjerg could not control his shot and found the side netting despite being allowed in on goal after a mix-up between Brighton defenders.
It was inevitable the equaliser would come and when Ings was set clear by Nathan Redmond's pass there was little doubt he would finish for his 20th league goal of the season.
Southampton defender Jannik Vestergaard was then inches away from making it 2-1 as his long-range shot bent viciously in the air but keeper Matt Ryan's fingertips pushed it onto the crossbar to keep Brighton in the contest.
A 76th minute breakaway offered the visitors a rare chance to restore their lead but Yves Bissouma missed from close in.