Mark Wright gave the visitors the lead after 13 minutes when he fired home after his initial stop was saved by the Iron keeper Joe Murphy.
United's best first-half chances fell to Martyn Woolford, whose shot was tipped over by Michel Kuipers, and a volleyed shot from Paul Hayes that went wide.
Jim McNulty doubled Brighton's advantage just after the break with an easy tap-in from Dean Cox's cross.
As Adkins rotated his players chances were hard to come by, with Woolford firing over and Hayes sending a shot into the side-netting after a corner.
The Iron are back in action on Tuesday night when the Premier League side Wolves visit Glanford Park.