The visitors appeared to be heading for a comfortable win when they scored three tries inside the opening 12 minutes, including two for Luke Gale.
However, London responded with four before the break to ensure that they led at half time.
The Broncos continued to dominate after the break, with Nick Slyney adding their fifth try to extend their advantage to 24 points.
Brett Kearney stopped that run just short of the hour mark before Gale completed his hat-trick to edge Bradford back in front.
Joshua Drinkwater dashed over for a second late on to keep the Broncos in contention, but Gale forced his way over for his fourth before a last-gasp effort from Manase Manuokafoa put the result beyond doubt for the Bulls.