Salford claimed their first win at Odsal for 23 years as Bradford slumped to their ninth straight defeat.
The Bulls made a start that has been typical of their recent struggles, falling two tries behind within the opening six minutes.
Jodie Broughton raced the length of the field to open the scoring before Ste Tyrer crept in at the corner, Daniel Holdsworth converting both.
Bradford hit back when Steve Menzies touched down Danny Addy's grubber kick but Stuart Littler collected a kick from Holdsworth for Salford's third.
Vinny Finigan stretched to score from Brett Kearney's grubber kick to mark his debut with a try and Addy's conversion closed the gap to eight at half-time.
But that platform was wasted as Salford were allowed to run in tries through Karl Fitzpatrick and Broughton to seemingly end the game as a contest.
The Bulls roared back, Stuart Reardon, Elliott Whitehead and Finigan all scoring as they threatened a remarkable comeback but ultimately, they had given themselves far too much to do.
Bulls: Vinny Finigan; Rikki Sheriffe, Michael Platt, Chris Nero, Stuart Reardon; Brett Kearney, Danny Addy; Andy Lynch, Heath L'Estrange, Craig Kopczak, Elliott Whitehead, Jamie Langley, Steve Menzies. Interchange: Wayne Godwin, Mike Worrincy, Steve Crossley, Glenn Hall.
Salford: Karl Fitzpatrick; Jodie Broughton, Mark Henry, Ashley Gibson, Ste Tyrer; Matty Smith, Daniel Holdsworth; Ray Cashmere, Malcolm Alker, Adam Neal, Ian Sibbit, Stuart Littler, Luke Swain. Interchange: Jeremy Smith, Adam Sidlow, Rob Parker, Luke Adamson.
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