The Bulls looked to have secured their place in the last 16 when they led 26-0 after 43 minutes but conceded seven tries in a remarkable 26-minute spell in the second half to crash to an ignominious defeat.
Steve McNamara's men looked a class apart in a one-sided first half as they aimed to maintain the only 100% record in Perpignan.
It looked straightforward for the visitors when veteran second rower Steve Menzies broke clear to send centre Paul Sykes away for a seventh-minute try and skipper Paul Deacon kicked the first of five goals to take his career points total beyond 2,500.
The Catalans, beaten finalists at Wembley two years ago, were in total disarray as Bradford ran in further tries through wingers Semi Tadulala and Rikki Sheriffe and second rower Matt Cook to establish a 20-0 lead after 21 minutes.
When Menzies added a fifth try early in the second half, it looked all over but the Dragons shook off their first-half lethargy to score four tries in a devastating 14-minute spell to cut the gap to just four points.
Prop Olivier Elima started the spree after a superb offload from skipper Greg Bird and other tries came from full-back Clint Greenshields, stand-off Thomas Bosc and Elima again as Bradford wilted in the warm sunshine.
Tadulala grabbed a second try after collecting Ben Jeffries' high kick to stretch the Bulls' lead but centre Jean Phillippe Baille and Bosc went over in quick succession to edge the home side in front for the first time.
The turning point came on 71 minutes when Bradford full-back Michael Platt offloaded from a three-man tackle behind his own line to gift a try to substitute Cyril Gossard.
Bosc's sixth goal put the French side eight points clear, which was a big enough cushion to enable them to withstand a late Bradford onslaught.
Substitute winger Dave Halley was awarded a penalty try after being tackled high as he went for the line but time eventually ran out for the Bulls.
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