Chris Hussey had put the Shakers in front on his full debut, but David Flitcroft's side could not hold on and Luke Norris grabbed the visitors' equaliser.
Both sides attacked with intent despite the poor conditions, Craig Jones heading an early chance over the crossbar for Bury.
Daggers midfielder Abu Ogogo also wasted a great opportunity, slicing wide after Medy Elito's corner had been flicked into his path.
The game opened up further after the break and Bury's Clive Platt was denied the chance of scoring in an empty net when Danny Nardiello's pass held up on the muddy turf.
But on-loan Burton defender Hussey latched on to Danny Mayor's pass to smash the ball past visiting goalkeeper Chris Lewington at his near post in the 60th minute.
Lewington then made a stunning one-handed save from Platt to keep the score at 1-0 and the visitors took full advantage.
Norris rounded off an impressive, jinking run with a composed finish to level the scores in the 67th minute.
Bury substitute Andy Procter came closest to a winner, heading against the crossbar from a Hussey corner.