Micky Taylor’s men went in at the break with the visitors 2-0 ahead, thanks to goals from Niall Cummins and Michael Elstone.
Dawson pulled one back in the second half, but despite piling the pressure on the hosts couldn’t find a leveller.
Both sides started the game quietly but with 13 minutes gone Cummins bundled the ball home after Brig failed to clear their lines.
With their noses already in front, it got even better for Marine just short of the half hour mark. Elstone’s cross into the box deceived the Brig defence and found its way into the top corner.
The home side were lacking fluency in their play in the half, but 10 minutes into the second half substitute Dawson dragged them back into the contest. After Patrick Wharton saved the original header, Dawson was first to react and found the back of the net from close range.
That goal spurred Taylor’s men on and they looked the livelier of the two teams, without conjuring up a whole host of chances. Brig probed and probed but Marine defended their lead excellently.
Deep into the second half Danny Forbes flashed a header just wide of the mark, which proved to be Brig’s best chance of salvaging a point.
On-loan forward Luca Navarro sent a shot wide in injury time - that the last piece of action as Marine took all three points at the Sir Tom Finney Stadium.