After holding their own in a scrappy contest they slammed home five perfect penalties to send the Bantams crashing to defeat.
After city midfielder Tom Kearney had missed their first penalty, the Quakers always had the initiative and striker Glen Robson struck the winner in front of Darlington's ecstatic fans.
After City's fightback to earn a draw on Saturday they should have been fired up to end a miserable run of three successive cup exits under boss Nicky Law, but it was Darlington to set the early pace.
Quakers keeper Michael Price didn't have a shot to save in the opening 45 minutes and his only scare came when a mishit backpass forced him to boot the ball clear at the feet of Michael Branch.
At the other end former England schoolboy Ashley Nicholls and Robson were proving lively customers for Bradfords' disjointed defence.
Bantams keeper Mark Paston had to push away a Robson drive and he rode his luck when a thundering header from Craig Liddle rattled his bar.
The Quakers skipper bravely blocked a volley from Dean Windass in a rare City attack and Paston was soon in action again blocking at the feet of Nicholls.
The Quakers made a bright start to the second half and it took vital blocks from Simon Francis and David Wetherall to prevent Paston being tested further.
As extra time loomed Robson blazed high and wide with four minutes to go and Bradford substitute Ben Muirhead saw his late effort fly a yard wide.
The former Manchester United man wasted a great chance at the start of extra time when his shot was well covered by Price, but the Quakers still looked most likely to break the deadlock.
Robson saw his glancing header drift wide of the target and it took a desperate block from Bradford full-back Gareth Edds to keep out a Barry Conlon shot.
Bradford: Paston, Edds, Wetherall, Francis, Heckingbottom, Gray, Kearney, Evans, Emanuel, Windass, Branch.
Subs: Combe, Wolleaston, Sanasy, Muirhead, Cornwall.
Darlington: Price, Liddle, Clarke, McGurk, Bossy, Nicholls, Keltie, Maddison, Valentine, Robson, Conlon.
Subs: Clark, Hughes, Waller, Sheeran, Norton.
Referee: Lee Mason (Lancashire).