The Bulls' second rower Sonny Nickle (fractured arm) and prop Brian McDermott (suspension) made comebacks, both scoring tries as Bradford clicked into gear at just the right time.
Leeds' chances of claiming third spot were dashed by half-time at The Stoop when they trailed 28-0 to a Broncos side that has now lost only one of their last seven Super League matches.
Paris St-Germain and Oldham, two clubs struggling at the foot of the table, were denied victories in frantic finishes against Warrington and Halifax.
An try in injury time by the Warrington hooker John Hough allowed Lee Briers to kick his sixth goal and condemn the Parisians to a 37-34 defeat. Paris scored seven tries to Wire's six but Jeremy Robinson converted just three of their eight goal-kicks.
Oldham's prospects were similarly dashed in the closing moments as Halifax's Australian threequarter, David Bouveng, scored his second try of the match and Martin Pearson added the touchline conversion with the final kick to secure a 30-29 victory.
Results and tables, page 19