The Seagulls were looking over their shoulders after three straight defeats and no wins in five meant they kicked off level on points with resurgent Fulham in 18th.
But victory at St Mary's, their fifth away from home this season, gives Graham Potter's side some breathing space at the bottom and, notably, saw them climb above Newcastle.
Adams could have put the hosts ahead with barely a minute on the clock but he was unable to get any power on his header from Ryan Bertrand's cross.
The former Birmingham striker then linked well with Kyle Walker-Peters and squared the ball to Stuart Armstrong, who shot straight at Robert Sanchez from 15 yards.
Fraser Forster, preferred to Alex McCarthy in Southampton's goal, made a fine save to palm a fierce Neal Maupay drive over the crossbar.
But the Scottish keeper was beaten from the resulting corner by centre-half Dunk's towering header.
However, Saints pulled themselves level when Dunk could only help Bertrand's header into the path of Adams, who was lurking alone at the far post to volley home.
Brighton should have taken the lead again before half-time when Dan Burn played Pascal Gross through on goal.
But as has so often been the way this season for the Seagulls, the German midfielder could not convert the chance, shooting too close to Forster who was able to deflect the ball wide.
This time it was not so costly, however, as Brighton managed to register a second goal in a Premier League match for the first time since the turn of the year with Trossard's stylish 56th-minute strike.
The Belgium winger barely had to break stride as he scythed through the Southampton backline and crashed his shot past Forster.
Southampton tried to hit back again and Seagulls keeper Sanchez saved with his legs from Adams before James Ward-Prowse's shot was charged down
But it is now 10 defeats from their last 12 matches for Southampton, who could really do with finding a win from somewhere just to ease any fears of being dragged into trouble themselves.