John Still's side, who have not won in seven league games, were on top for long periods but could not convert their pressure into goals as Peter Thorne, David Wetherall and Nicky Law (2) lifted Stuart McCall's City clear of the relegation dogfight.
The hosts were in command for the opening 20 minutes, but Thorne once cost Cardiff £1.8million and he showed why with a predatory finish after Bradford had survived.
Set-pieces had looked the most likely source of an opening for the visitors, and McCall's side had already seen Paul Heckingbottom have a goal ruled out for pushing at the far post.
But the full-back tried again and when his inswinging free-kick was unwittingly flicked on by Daggers' defender Danny Foster, Thorne was on hand to head home.
Despite having won 15 corners by the hour mark the Daggers couldn't equalise and their hopes were dealt a blow when referee Philip Joslin sent-off Soloman Taiwo for what he thought was a two-footed challenge on Darren Williams.
The decision was harsh in the extreme and provoked outrage from the home fans.
Their anger was increased when Wetherall scored with 15 minutes to go, heading in another Heckingbottom set-piece.
Paul Benson - scorer of 28 league goals last term - pulled one back with a smart low finish from another set-piece and only two superb saves from Donovan Ricketts kept Bradford in front.
Substitute Law made the most of the man advantage by slaloming through the hosts' rearguard in injury-time to add an undeserved gloss to the scoreline, leaving Bradford looking up at mid-table and Still's Daggers worrying about an immediate return to the Conference.