A hat-trick of set-pieces helped Darlington to come back from a goal down and give them their first FA Cup win since November 2010.
They will be at home to Blyth in the next round, though a dat is yet to be decided as Bishop Auckland are also at home in the next round on September 13.
Most of the early pressure belonged to Quakers, but they didn’t create a chance until the tenth minute, when Terry Galbraith moved forward and hit a deflected 30-yarder into the hands of keeper Dan Lowson.
West had a penalty shout for handball turned down when Dennis Knight’s shot from just inside the box was blocked by a defender.
Quakers went close on 16 minutes when Adam Mitchell, with hardly any backlift, struck a dipping right foot shot just over the bar.
But at the other end Robbie Briggs opened up the home defence with a run through the middle, but from his lay off Steve Richardson side-footed tamely at Peter Jameson.
Lowson pulled off a good save on 38 minutes when Stephen Thompson, who had switched wings, pulled the ball back from the right for Mitchell to hit first time, but the former Darlington keeper saved low down.
West were awarded a penalty on 39 minutes when Knight turned quickly inside the Darlington box and was tripped by Galbraith, and the former Newcastle youngster tucked the spot kick into the bottom left hand corner.
The game became fractious at the start of the second half, and Knight was booked for deliberate handball as was Quakers’ Gary Brown for a foul as the game was punctuated with stoppages.
West nearly went 2-0 up from a quickly taken free kick that ended with Jameson fumbling a Ward cross, and Brown cleared Richardson’s follow up off the line. That was a turning point, because Quakers levelled on the hour.
Lowson, who had just received lengthy treatment, pushed a low shot from Dowson around the post. Galbraith curled a teasing corner into the six yard and West defender Darryl Hall turned the ball past his own keeper despite being under no pressure.
Lowson pulled off a superb save a minute later, when Thompson crossed for Dowson to head towards goal, the keeper going full length to tip the ball over the bar.
Quakers took the lead on 69 minutes from another set-piece, this time from the opposite side. Mitchell curled the ball to the corner of the six yard box, to where Armstrong timed his run perfectly and headed past Lowson off the inside of the post.
Quakers then became guilty several times of being caught in possession, and on one occasion Knight dispossessed Brown and played a one-two with Richardson, before firing over.
Quakers made sure five minutes from the end, when Mitchell swung a corner over to the far post, where Liam Hatch, who had only been on the field for a few seconds, headed into the top corner and gave Darlington their first FA Cup replay win since 1998.
Manager Martin Gray said: “My half time team talk was a big factor, because West controlled the first half. I said a few words at half time because I wasn’t happy.
“It was good to score from set-pieces, we’ve done well from those so far this season.”
Goals: Knight (0-1); Hall OG (1-1), Armstrong (2-1), Hatch (3-1)
Referee: Gareth Rhodes (Selby)
Darlington: Jameson, Brown, Galbraith, White, Hunter, Portas, A Mitchell (Fisher 88), Scott (Robinson 70), Armstrong, Dowson, Thompson (Hatch 83). Subs (not used): Bell (gk), Reed, Purewal, Young
West Auckland: Lowson, Gibson, Galpin, Close, Wilkinson, Hall, Richardson, Francis, Ward, Knight, Briggs. Subs (not used): Vipond, Byrne, Ayers