The scoreline ends a winless run that stretched to eight games for Jamie Milligan’s side, and re-ignites their push for the play-off places in the NPL Premier Division.
It was Brig who had the first real sight at goal, with Sinclair-Smith forcing a good save out of Grantham goalkeeper James Pollard.
The visitors, however, got the opening goal of the contest on 27 minutes.
After a goalmouth scramble, Steven Leslie managed to divert the ball over the line from close range.
After more pressure, Brig got their leveller on 36 minutes. Sinclair-Smith ran through on goal and diverted the ball beyond Pollard.
Dawson almost gave the hosts the lead shortly before half-time, heading over the bar from Chris Churchman’s cross.
The second half started at a fast tempo, with Grantham coming close when Harry Wood saw his effort go over the bar.
But it was Brig who made their chance count. On 56 minutes, Dawson received the ball inside the box before shooting beyond Pollard to give Milligan’s side the advantage.
Brig continued to have opportunities at goal, with Pollard forced into a fine save to deny Isaac Sinclair’s header.
After five minutes of stoppage time at the end of the second half, the hosts managed to hold on to their advantage to gain a crucial three points.
The result takes Brig up to seventh place in the league table, and they are now six points off the play-off places.