Bradbury, once a £3.5million striker at Manchester City, is now playing at right-back.
From that position Bradbury cut in from the touchline in the 24th minute to beat goalkeeper Tony Roberts with a deflected left-foot shot.
Bournemouth added a second goal in the 31st minute when Brett Pitman netted his fourth of the season with a near-post run to turn a centre from Danny Hollands past Roberts from six yards.
Hollands almost got on the scoresheet two minutes later with a left-foot shot which was superbly blocked by Roberts.
Dagenham threatened little in the first half although an early shot from Ben Strevens was pushed away at full stretch by home keeper Shwan Jalal.
Manager John Still made two substitutions in the 35th minute and his third in the 69th gleaned a goal three minutes later.
Danny Foster took a free-kick on the right and Jon Nurse, the final substitute, rose about the defence to head home.
The lively Nurse almost equalised in the 84th minute when a fierce left-foot shot struck the angle of bar and post.
Dagenham put together some pressure towards the end of the match, but Bournemouth held on for their third win of the season.
Cherries substitute Josh McQuoid should have added a third in injury time, but fatally halted when clean through, thinking a free-kick had been awarded and by the time he had regained control of the ball, the opportunity was lost.
Bournemouth had striker Jeff Goulding and hard-working Marvin Bartley booked as Dagenham threatened to equalise in a hectic spell towards the end in front of Dean Court's lowest crowd of the season.