The Shakers were low on confidence after four straight league defeats and it showed early on with a nervy visiting defence giving Daggers full-back Scott Griffiths time to hammer a volley inches wide in the 12th minute.
The visitors weathered the storm and caught the home side with a sucker-punch in the 20th minute when Paul Scott poked the ball home from three yards after a fumble by goalkeeper Tony Roberts.
The home side tried to claw their way back into the game with midfielder Glen Southam hammering a speculative 25-yard strike high and wide of the goal on the half-hour.
Daggers striker Ben Strevens went even closer in the 36th minute when he fired a spectacular bicycle kick inches over Bury goalkeeper Jim Provett's goal from six yards.
The home side were rewarded for their efforts in the 39th minute when midfielder David Rainford rose highest in the Bury box to power a Southam cross pass Provett from six yards.
Daggers came flying out of the blocks in the second period with striker Chris Moore going close with a volley from ten yards in the 51st minute.
The home side should have taken the lead in the 70th minute when Shane Huke flashed a powerful 25-yard strike narrowly wide.
Strevens should have done better moments later when he hacked the ball over the bar from six yards after shaking off Bury skipper Dave Challinor inside the box.
Chris Casper's side held on to deny the home side a winner and came close to snatching an unlikely victory in the 88th minute when substitute striker Andy Mangan fired against the crossbar from the edge of the area.