Wayne Burnett's Daggers remain unbeaten at home this season and are now without defeat in their last five games thanks to goals from Afolabi Obafemi and Rhys Murphy.
The visitors had enjoyed the better of the earlier chances, starting with Craig Jones who headed straight at Chris Lewington, and it was against the run of play when the hosts took the lead two minutes before the break.
Zavon Hines slipped the ball to 18-year-old Obafemi, and he skipped inside Gareth Roberts before smashing a superb left-footed shot past Shakers goalkeeper Trevor Carson to register his first senior goal.
Dagenham doubled their lead shortly after the restart when Murphy latched onto Billy Bingham's free-kick before slotting comfortably past Carson.
Minutes later Murphy almost grabbed his second yet Carson superbly saved his free-kick.
Anton Forrester pulled a goal back for Bury with an 89th-minute penalty after Bingham had handled in the box.
Roberts then hit the post in stoppage-time as the visitors almost levelled.