Dagenham condemned Bury to a fourth straight defeat in all competitions after two second-half goals.
The Daggers took the lead when debutant Alex Jakubiak poked home after a knockdown inside the six-yard box.
The hosts created most of the chances throughout the afternoon, with strikers Danny Rose and Daniel Nardiello both having goals ruled out for offside.
Jamie Cureton secured three points when a static defence allowed him to finish smartly from inside the area.
Shakers boss David Flitcroft responded to his side losing three matches in a row in all competitions by making five changes, including handing debuts to on-loan signings Adam El-Abd and Robert Milsom.
He watched his team create numerous chances throughout, with Craig Jones, Rose and Nardiello all going close.
However, Wayne Burnett's side always looked dangerous on the break as he played three up front and were more clinical with their opportunities.
Dagenham & Redbridge first team Coach Darren Currie speaks to BBC London 94.9:
Darren Currie: Daggers coach on Bury win
"We started brightly and got a few decisions and an early corner and started on the front foot.
"Bury are a good side and they pinned us back a bit and they probably had the better of the first half.
"We are away from home and the chances are few and far between and when they come along you have to make them count and we have done that today."
Bury manager David Flitcroft told BBC Radio Manchester:
Flitcroft on Bury v Dagenham & Redbridge
"I'm not too sure where the game got away from us. It seemed to suck the life out of us, the goals that Dagenham scored.
"But still, we created so many chances to win this football match.
"Dagenham have thrown their bodies on the line today and it was a really spirited performance, but we have to find a way to be clinical and ruthless.
"Our end quality wasn't there and our end game wasn't on, but that has been a trend over the last few games."
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