The visitors produced most of the first half opportunities without really threatening the Quakers` goal.
Home goalkeeper James Talbot was forced into a decent save from new signing Tom Walker, but the opening 45 minutes finished goalless.
Early in the second period, Darlington were awarded a penalty when Greg Mills was tripped in the box by Walker.
However, Reece Styche tried to be a little too clever, attempting a mid-goal chip which Ammies` keeper Max Crocombe simply stood still and caught with ease.
It was an embarrassing moment that would come back to haunt the player later.
And almost inevitably, Salford went ahead and broke the deadlock shortly afterwards in the 54th minute when a free-kick was set back to Mike Phenix, who hammered the ball past Talbot.
But on 65 minutes, Darlington equalised, with Styche having a hand in the goal that Mills scored from close range after being set-up by Josh Gillies.
Both sides had chances to go ahead after that, with Quakers` Phil Turnbull clearing one off the line.
And then, with less than a minute of normal time left to play, the away end erupted as a long ball over the top found Anthony Dudley, who raced through to beat Talbot and scored a crucial winner for the visitors.