Captain Simon Ainge scored the only goal 10 minutes from time, with Brig unable to find what would have been a deserved equaliser in the closing stages.
The build up to the game was dominated by the weather, with heavy deluges of rain before kick-off, although by the time the match began the sky had cleared somewhat.
Brig began brightly, Adam O'Reilly hitting an effort well from distance but narrowly clearing the bar.
Matt Hudson then made two excellent one-on-one saves from Simon Ainge and Luke Lofts, before visiting goalkeeper Tom Jackson parried away a Rowan Roache volley which looked destined for the bottom corner.
Isaac Sinclair then took a Danny Forbes cross-field ball in his stride and forced a fingertip save from the Gainsborough stopper.
Sinclair then had another half chance on the stroke of half-time after being played through by Roache, but Jackson rushed out quickly and prevented him from getting a clear shot away.
Brig took their positive momentum into the second half and were inches away from taking the lead on 57 minutes, Forbes striking the post with a right-footed curling effort. Unfortunately for Jamie Milligan's side, the ball then bounced clear from danger.
They then hit the post again 10 minutes later, Macauley Wilson's long distance effort evading everyone but coming back off the woodwork.
A subsequent corner then fell to debutant Matthew Flynn, whose half-volley was tipped over by Jackson.
Despite the Brig pressure, however, it was Gainsborough who took the lead when Ainge lifted a rebound into the bottom corner following a good initial save from Hudson.
Brig continued to press forward in the closing stages in an attempt to score what would have been a deserved goal, but the chance did not arrive and the game ended in a 1-0 victory for the visitors.