Gratton scored his goal from the penalty spot mid-way through the first half before Greaves sealed three points for the Seasiders with a spectacular lob as Town looked to build some momentum in the Pitching In Northern Premier League Premier Division, writes Liam Ryder.
The return of Connor Smith to Marske United at the end of his loan spell saw Soni Fergus return to the starting XI, while Aaron Haswell and Connor Simpson came in for Bradley Fewster and Greaves who had both started the previous weekend's defeat at Macclesfield.
Town, playing their first home league game in almost two months, came close to opening the scoring in the second minute when Simpson took advantage of some loose play in the visitor's defence. He struck towards goal but James Pradic made a routine save, gathering the ball with two hands.
After that let-off for Brig, the Seasiders continued to push for an opener and in the tenth minute, Junior Mondal's corner caused problems for the Lancashire outfit's defence and the ball once again found its way to Simpson, but he could only head into the hands of Pradic.
But the visiting goalkeeper was unable to do anything 15 minutes later as Gratton opened the scoring.
The Town midfield swarmed to gain possession and the ball was slipped into the path of Lewis Hawkins. Hawkins took the ball beyond one man before being fouled inside the area.
Gratton stepped up to send Pradic the wrong way and set the Seasiders on their way on a wet afternoon at the Turnbull Ground.
In the 32nd minute, Brig had their first real sight of goal as Jack Baxter hit the target having bent a free-kick around the wall. Shane Bland was able to smother the ball and kill off any danger for his side.
Town kept on pushing for a second and it almost arrived in the 38th minute as Hawkins, eventually given the Player of the Match Award, looked up and saw an opportunity to strike for goal from distance. Hawkins came close but Pradic happily watched the ball sail past his left-hand post.
In the second half the conditions worsened as heavy rain lashed down onto the Turnbull surface, creating difficult conditions for both sets of players.
But chances still came around and Junior Mondal shot straight at Pradic in the 80th minute before, eight minutes later, substitute Greaves headed narrowly wide following an inch-perfect cross from Alfie Doherty.
Greaves wouldn't be denied moments later, however, as he beat the offside trap before calmly finding the net with a chipped effort over the on-rushing Pradic just before five minutes of stoppage time were announced.
And Nathan Haslam's side would see out the victory and claim a first clean in the league sheet since the opening day of the season as they prepare for a trip to the league leaders, Radcliffe, on Tuesday night.