James Hanson headed home from Robbie Threlfall's corner after just two minutes but City failed to make the most of their first-half dominance with striker Gareth Evans and winger Omar Daley wasting chances.
Goalkeeper Matt Glennon made a crucial block at the start of the second half to deny Daggers captain Mark Arber, but Hanson's luck was out when his header hit the post with Tony Roberts flat-footed.
Daley limped out of the action after pulling a hamstring, but that looked like a minor setback when Hanson got on the end of Threlfall's 67th-minute free-kick to bag his second.
The two-goal lead was short-lived however as substitute Jon Nurse lashed home a fierce drive and when City centre-half Steve Willams diverted a loose ball into his own net with 15 minutes left, the game was up for grabs.
Arber survived a strong penalty shout for handball, but the Bantams edged ahead again in the 82nd minute when on-loan Hull striker Ryan Kendall coolly lobbed Roberts from 14 yards to notch his first league goal.
More sloppy defending gifted possession to Daggers winger Danny Green and Bantams centre-half Luke Oliver watched in horror as his diving header arrowed into his own net.
The visitors finished strongly and City were lucky that neither Paul Benson nor Phil Walsh converted chances.
Bradford City: Glennon, Rehman, Williams, Oliver, Threlfall (J O'Brien 90); Daley (Kendall 60), Flynn (O'Leary 67), L O'Brien; Hanson, Evans. Unused substitutes: McLaughlin, Bateson, Osborne, Horne.
Dagenham & Redbridge: Roberts, Ogogo, Doe, Arber, McCrory; Green (Currie 90), Pack, Folly (Nurse 65), Gain: Scott (Walsh 82), Benson. Unused substitutes: Lewington, Uddin, Montgomery, Tejan-Sie.
Referee: P Tierney (Wigan).
Man of the match: James Hanson.
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