The first real chance fell to the visitors in the 11th minute, when Danny Carlton leapt highest to nod a Franz Burgmeier corner inches over the bar.
Darlington midfielder Jason Kennedy blasted a 25-yard strike well wide six minutes later after dribbling through midfield.
Paul Benson thought he had given Daggers the lead in the 25th minute when he fired in from close range - only for the linesman to raise his flag for offside.
The striker had another effort ruled out just two minutes later when he powered in a bullet header from an offside position.
The Quakers almost took the lead against the run of play in the 33rd minute when Ricky Ravenhill unleashed a thunderous strike from the edge of the box which Daggers keeper Tony Roberts did well to tip around the post.
Jon Nurse wasted a gilt-edged chance to put Dagenham ahead on the stroke of half-time, when he ran onto a clever Benson lay-off on the edge of the area before firing high and wide with the goal at his mercy.
Both sides struggled to find their rhythm after the break, with Darlington's Rob Purdie volleying a dipping strike over the bar from 20 yards in the 67th minute.
Ravenhill should have done better in the 71st minute when he side-footed over the bar from close range after being teed up by Carlton.
Purdie missed another great chance two minutes later when he blasted over when one on one with Roberts.
The visitors finally broke the deadlock in the 74th minute when Pawel Abbott ran on to a clever Stephen Foster throughball before slotting calmly under the onrushing Roberts to give his side the lead.
Matt Ritchie came close to snatching a late equaliser when he skimmed the bar with a superb volleyed effort in the 88th minute but The Quakers held on for a valuable three points.