The Bulls led 20-0 at half time in Perpignan and the game looked over as a contest.
However, forty second-half points from the Dragons gave them a two-point victory over Bradford and a place in the next round.
But the Catalans quietly began to work their way back into the game: tries from Olivier Elima (2), Clint Greenshields and Thomas Bosc, who also kicked three goals, to make the score 26-22 to Bradford.
Semi Tadulala got over for Bradford to extend the lead but the Dragons comeback was irresistible, Philippe Jean-Baile and Cyril Gossard, Bosc chipping in with another try and three goals to take a 40-32 lead.
A late try from Dave Halley, Deacon converting to reduce the deficit to a mere two points, gave the Bulls hope of a comeback of their own, but the Catalans had done enough to claim a remarkable victory.
Catalans – 40
T: Elima (46, 56); Greenshields (48); Bosc (52, 65); Jean-Baile (61); Gossard (71)
G: Bosc (46, 53, 56, 61, 65, 71)
Bradford – 38
T: Sykes (7); Sheriffe (10); Cook (19); Nero (21); Menzies (41); Tadulala (59); Halley (73)
G: Deacon (7, 19, 44, 59, 74)
Ref: S Ganson