Bradford ended their five-match losing run with a hard-fought goalless draw.
City dominated large chunks of the game but solid Darlington defending and wayward finishing cost them victory.
An unmarked Barry Conlon headed a first-half Paul Heckingbottom free-kick wide Quakers keeper Andy Oakes kept out a fierce shot from Omar Daley.
Free-scoring Darlington created few chances and Bradford right-back Darren Williams blocked their best effort - a 55th-minute shot from Greg Blundell.