Mistakes by Peter Gain and Tony Roberts in the first half set the Shakers on their way, before Elliott Bennett sealed the three points with a classy late goal.
The Daggers started brightly with striker Ben Strevens nearly putting his side ahead in the 14th minute when he climbed highest in the Bury box to meet a Saunders corner only to see his bullet header saved on the line by keeper Wayne Brown.
Daggers striker Paul Benson took Bury full-back David Buchanan out of the game with an audacious lob in the 21st minute but fired a powerful strike high over the bar from 15 yards.
Bury caught the home side with a sucker punch in the 32nd minute when Stephen Dawson pounced on a mistake by Gain and clipped the ball over keeper Tony Roberts into the net.
Gain nearly made up for his error three minutes later when he unleashed a thunderous shot from 25 yards which Brown did well to palm to safety.
The Daggers were still rocking from the first goal and Bury took full advantage in the 36th minute when Mike Jones caught Roberts knapping with a speculative 35-yard effort which flew over the goalkeeper's head into the back of the net.
Strevens should have snatched one back for the home side on the stroke of half-time when he stabbed the ball straight at Brown from point-blank range after racing onto a superb Paul Benson cross.
Bury looked lively after the break with Bennett firing narrowly over in the 55th minute after pouncing on a loose ball on the edge of the Daggers box.
The Daggers left gaps at the back as they searched for a late goal, allowing Bennett to find the space to fire a third goal past Roberts from the edge of the area to seal the points for Bury in the 83rd minute.
Sam Saunders curled in an impressive late consolation goal for the Daggers in stoppage time but there was no way back for the home side.