Matt Wolfenden gave the hosts the perfect start with a goal after two minutes but Kieran Charnock levelled five minutes later.
On the stroke of half-time, Celtic went in front again through Scott Bakkor.
And it was 3-1 in the 65th minute when Neil Kengni found the net.
Two minutes later Ryan White reduced the arrears but Brig could not find the equaliser.
Elsewhere, Lancaster City were held to a goalless draw at Giant Axe by North Ferriby United.
In the NPL First Division West, Clitheroe went down 2-0 at home to Colwyn Bay, while Kendal Town lost by the same scoreline at Mossley.
AFC Fylde climbed to fifth place in the National League after a 3-0 home victory over Aldershot Town.
Southport kept a clean sheet in National League North for the first time this season, drawing 0-0 at home to Darlington.
But for the second match in succession they missed a penalty against a mediocre side that would have given them their first home win since March.
On another day they might have had five goals, with Bradley Bauress hitting the crossbar after 11 minutes and Dion Charles, Davern Green and David Morgan all having shots cleared off the line.
The penalty was awarded in the 56th minute when Jack Sampson was pulled down. Bradley Bauress’s spot kick was saved by Jonny Maddison.
In the NWCL Premier Division, Jacob Salisbury scored both goals as Garstang won 2-1 at AFC Darwen. Charnock Richard were 4-1 winners at Penrith, while Burscough lost 4-2 at Irlam.