Andrew Barcham's goal after five minutes, a curled shot from the edge of the area, was enough to seal victory at Northolme.
Jimmy Ryan (groin) and Eddie Nolan (Achilles) sat out the match due to injuries.
There were few clear-cut chances for a second goal in the match, with Bobby Grant going closest for the Iron through a volleyed effort which was turned behind by Phil Barnes than a free-kick which was parried away by Andy Pettinger, who had replaced Barnes in the home goal.
Trinity missed the opportunity to level in the closing stages
when Zac Delcompagni fired wide after rounding the Iron goalkeeper Josh Lillis.