The Bulls snatched defeat from the jaws of victory after gifting Hull FC both points at the KC Stadium.
Bradford did enough to win the game but errors proved costly as the hosts alleviated some of the pressure on under-fire boss Richard Agar.
Early tries from Paul Sykes and Shad Royston, both converted by Patrick Ah Van, allowed the visitors to open a 12-0 lead but they failed to defend it.
Instead, Kirk Yeaman burst through on the left to cut the gap and Danny Tickle added the conversion before scoring himself and adding a further goal.
Bradford edged back in front after good work by Marc Herbert allowed Kearney to send in Chev Walker, Ah Van converting.
But Hull levelled on the stroke of half-time when former Bull Lee Radford cut an angled line through the defence, Tickle improving.
The hosts went ahead five minutes after the interval when Richard Horne's break set the platform for Jordan Turner to score, Tickle adding the extras.
But, despite Ian Sibbit being sent to the sin-bin, Royston scored off Herbert's kick to level once more, Ah Van converting.
Herbert missed a drop goal and when Tickle attempted one of his own, Hull enjoyed a slice of luck, the kick scuffed across field, where Tom Briscoe scored, Tickle improving.
Briscoe scored again late on when Bradford were caught chasing the game.
Bulls: Brett Kearney; Shaun Ainscough, Chev Walker, Shad Royston, Patrick Ah Van; Paul Sykes, Marc Herbert; Andy Lynch, Matt Diskin, Bryn Hargreaves, Elliott Whitehead, Tom Olbison, James Donaldson. Interchange: Heath L'Estrange, Tom Burgess, Craig Kopczak, Ian Sibbit.
Hull FC: Cameron Phelps; Will Sharp, Jordan Turner, Kirk Yeaman, Tom Briscoe; Richard Horne, Sam Obst; Mark O'Meley, Danny Houghton, Sam Moa, Willie Manu, Danny Tickle, Craig Fitzgibbon. Interchange: Epalahame Lauaki, Joe Westerman, Richard Whiting, Lee Radford.
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