Diego De Girolamo netted twice for the visitors, with Warren Clarke and Jason Gilchrist also getting their names on the scoresheet.
The game started at a fast pace, with both sides having early chances. James Cottam made a good early save to deny a deflected De Girolamo shot. At the other end, Isaac Sinclair’s superb low cross narrowly evaded everyone at the far post.
Brig came closest to opening the scoring on 25 minutes. Sinclair found space in the box and fired an effort towards goal, but it hit the post and bounced clear.
From the resulting counter attack, Buxton opened the scoring. They worked the ball to De Girolamo, who managed to squeeze the ball beyond Cottam from close range.
There was little to report on for the remainder of the first half, with the visitors entering half time with a narrow 1-0 lead.
Two minutes into the second half, however, Buxton extended their lead. Clarke met a low cross at the near post and flicked the ball beyond Cottam.
It was 3-0 on 51 minutes. De Girolamo found space before firing a low effort into the bottom corner from 18 yards.
Buxton then added a fourth goal on 79 minutes, with Jason Gilchrist finding the back of the net from close range after being played through on goal.
Brig had a couple of late chances to get themselves on the scoresheet. Jake Thompson forced a save out of Buxton goalkeeper Theo Richardson, whilst Macauley Wilson had a close range effort cleared off the line. But a goal was not forthcoming, and it was the visitors who left Sir Tom Finney Stadium with all three points.
Brig will look to get back to winning ways in the NPL Premier Division on Tuesday when they travel to Atherton Collieries.