Craig Carney, Rob Apter and Paul Dawson all struck in the dying moments to turn one point into three on the road.
Brig had taken a 50th-minute lead through Preston North End loan striker Joe Rodwell-Grant, only for Callum Saunders to equalise seven minutes later for the home side.
And it stayed that way until the 89th minute when Carney’s strike gave Brig the lead again late in the day.
That was immediately followed by Apter getting a third goal for the visitors and in stoppage time, Dawson added the gloss with a fourth.
Lewis Fensome got the only goal as Lancaster City won 1-0 at home to Buxton, elsewhere in the division.
And in the NPL West Division, Kendal were 2-1 winners at home to Market Drayton Town, while Clitheroe won 1-0 at Newcastle Town, Harrison Burke on target.
In the National League North, two goals from Joe Piggott were enough to give AFC Fylde a 2-1 victory over Telford United at New Bucks Head.
And Southport drew 1-1 at Leamington, Marcus Carver with the Sandgrounders’ equaliser.
In the NWCL Premier Division, Jordan Darr scored a hat-trick and Brad Carsley got the other two as Charnock Richard beat Litherland REMYCA 5-2.
But Moses Yoak’s goal was no more than a consolation for Longridge Town, as they lost 3-1 at home to Skelmersdale United.
Daniel Mitchley and Tom Croughan (2) got the visitors’ goals.
In the NWCL First Division North, Garstang came from 2-0 down to draw 2-2 at Campion.
A goal from Mark Ferguson and a Nicky Boshell penalty put the hosts in the driving seat. But Jacob Salisbury got one back before half-time and Joe Noblet’s 72nd-minute penalty rescued a point for the Riversiders.
In the FA Vase second round, Burscough went down 3-1 at Pilkington.