The Boro were twice pegged back by a plucky Brig side who looked set to earn an impressive point until Josh Hancock converted from the penalty spot - his second of the game - with 79 minutes gone.
Second-placed Warrington Rylands were indebted to another late goal by Adama Sidibeh, meanwhile, as they rescued a late point at home to Basford United.
The Lions had led through Remaye Campbell's goal shortly after the hour mark, but they remain in the bottom three following Saturday's draw.
Worksop Town also dropped points as they fell to a surprise 1-0 defeat to Marine.
Joseph Ferguson scored the only goal of the game for the Mariners, who were the only team in the NPL Premier Division to keep a clean sheet on Saturday.
The Tigers' first slip-up at home in the league in 19 months allowed Macclesfield to climb to third after a 5-1 victory at home to Guiseley.
The visitors had led through Kallan Murphy's 17th minute header, but the Silkmen came roaring back as five goals from five different scorers earned them a fifth win in six matches under new boss Alex Bruce.
Elsewhere, goals in either half from Harry Bunn and Jack Redshaw helped Hyde United to a 2-1 win against Morpeth Town, while Ashton United came from behind to beat Atherton Collieries 2-1 on the road. Colls have now lost 10 of their last 11 matches in all competitions.
A Karl Demidh double helped Matlock Town to a 3-2 win at FC United of Manchester, who remain 16th in the league table.
There were also wins for Ilkeston Town and Marske United, who overcame Gainsborough Trinity 3-2 to climb out of the relegation places.
The Seasiders leapfrogged Bradford Park Avenue, who were twice pegged back by Lancaster City, who earned a 2-2 draw in their first outing since the departure of manager Mark Fell.
Finally, Workington loanee Joe Bunney was on target for his new club for the first time as they left it late to rescue a point at Whitby Town, who were unable to enjoy a perfect week after defeating leaders Radcliffe in midweek.
Results - Saturday 25 November
Atherton Collieries 1-2 Ashton United
FC United of Manchester 2-3 Matlock Town
Hyde United 2-1 Morpeth Town
Ilkeston Town 3-1 Stafford Rangers
Lancaster City 2-2 Bradford Park Avenue
Macclesfield 5-1 Guiseley
Marske United 3-2 Gainsborough Trinity
Radcliffe 3-2 Bamber Bridge
Warrington Rylands 1-1 Basford United
Whitby Town 1-1 Workington
Worksop Town 0-1 Marine
Photo credit: Radcliffe