Callum King-Harmes gave them the lead in the first half, before Lewis McKinney equalised on the stroke of half-time.
Substitute Matt Dudley restored Brig's advantage, before Will Shepherd scored late on to give the visitors a share of the spoils.
Brig started the brightest of the two sides, with Danny Wisdom heading straight at Max Williams in the Salford goal and King-Harmes placing an effort narrowly wide.
Williams was then out quickly to get in the way of Wisdom's effort, before King-Harmes forced a fine parry out of the visiting 'keeper.
The Bolton loanee wasn’t to be denied again, however, on 31 minutes, this time making no mistake with a cool finish into the bottom corner following an excellent through ball from Scott Burton.
McKinney then equalised on 43 minutes, meaning both sides went into the interval on level terms.
Jamie Milligan made a total of five changes to his side at half-time, and it was Matt Dudley who scored within just eight minutes of coming on, emphatically slamming the ball into the corner to restore Brig's advantage.
Danny Forbes then had a couple of chances, but was well denied by sub goalkeeper Brad Rose on both occasions.
With just five minutes left on the clock, Salford equalised through Shepherd, the number 10 firing into the bottom left corner.
Brig pressed late on to try and retake the lead, but Forbes and then new signing Isaac Sinclair were both denied by the goalkeeper, with the game ending in a 2-2 draw.
Match photos: photo gallery available here Bamber Bridge v Salford City photos