Quakers are now unbeaten in five matches thanks a second half winner from David Syers.
They were a deserved three points against a play off chasing side for Quakers, who played with plenty of confidence and determination, but the only downside of the afternoon was a red card for Stephen Thompson.
But nevertheless, Quakers are now 14th in the table, five points above the relegation zone.
Bradford had the first opening after 3 minutes, a low right foot effort by Wayne Brooksby from the corner of the box that Pears got down comfortably to save.
Reece Styche then had a goal disallowed when he chipped the keeper, but he was pulled up for pushing defender Shane Killock as he broke through.
The game was mostly confined to midfield after that, with Styche working hard up front for Quakers.
Bradford won a corner off Galbraith on 17 minutes, that Luca Havern headed over from six yards.
Quakers had their first chance on 22 minutes when Thompson turned on a right wing throw from Luke Trotman, ran into the box, and fired left footed over the top.
Quakers had a good chance on 36 minutes, when David Syers nodded down a Phil Turnbull free kick from the left for Styche, with his back to goal, to volley over the top.
Thompson then picked the ball up in midfield from Styche, ran forward, and struck a right foot shot a couple of yards wide of the left hand post.
Thompson then burst into the box and pulled the ball back for Styche, but the striker took a touch and was crowded out.
Quakers had got the better of the last 20 minutes of the half, and had created more chances.
Bradford PA 0 Darlington 0
Quakers had a great chance at the start of the second half, when Wheatley was fouled 30 yards out, and Galbraith floated the free kick into the middle where Styche, at full stretch, sidefooted wide.
Quakers took the lead on 56 minutes. Thompson again burst down the right, got to the bye line, and pulled the ball back for David Syers to slot past Drench, his second in successive games and sixth of the season.
Syers nearly got a second, this time with a glancing header from an inswinging Galbraith free kick that went a yard over the top.
And another chance came Quakers’ way when Trotman crossed to the far post for Gillies to come in and see a stooping header cleared off the line.
But BPA replied with a move up the left that ended with Spencer firing straight at Pears.
BPA had more of the pressure in the closing stages , and Pears pulled off an excellent save from a Havern piledriver from ten yards.
Quakers were reduced to ten men on 85 minutes when Thompson was shown his second yellow card. He had been harshly booked for a foul a few minutes before, and after he went down when he was challenged, the referee showed him a second yellow for simulation.
BPA threw everything at Quakers, with Adam Nowakowski havin ga header saved, but Quakers held on for their third successive win.
Darlington: Pears, Trotman, O’Hanlon, Turnbull, Heaton, Galbraith, Gillies (sub Brown 84), Wheatley, Styche (sub Mills 89) , Syers, Thompson
Subs: Scott, Bancroft, Saunders
Bookings: Heaton, Styche, Syers
Red Card: Thompson
Drench, Toulson (sub Mulhern 68), Clee, Wroe, Killock, Havern, Brooksby (sub Vidal 51), Pybus (sub Nowakowski 77), Boyes, Johnson, Spencer.
Subs: Hill, Smith
Booking: Killock, Wroe