Leeds inflicted third-place Catalans' first home defeat since March 8 and kept the pressure on St Helens at the top of the engage Super League.
Behind at the break, the Rhinos produced a tremendous fightback thanks to the outstanding Ali Lauitiiti that ended the French side's seven-match unbeaten run.
Scott Donald's magnificent try capped the display as the Rhinos went back level with Saints.
Leeds opened the scoring thanks to Lee Smith, with Kevin Sinfield scoring the conversion.
Soon the Dragons began to show why they have been on their unbeaten run - the guile of halves Casey McGuire and Thomas Bosc perfectly complementing the drive of back rowers Jason Croker and Gregory Mounis.
After Ryan Hall had knocked on around his own 20, Catalans pounced from the scrum, Bosc and Clint Greenshields sending in Cyril Stacul at the corner.
Bosc landed a superb touchline conversion to level in the seventh minute.
Five minutes later, Hall was again exposed when McGuire's towering bomb eluded him and Olivier Elima collected to scamper over for another Bosc-converted try.
With the home defence dominating and Remi Casty, off the bench, producing a series of surging runs, Dragons extended their lead in the 25th minute.
Sebastien Raguin powered over out wide from Bosc's pass, the same man again landing a magnificent touchline conversion.
Leeds cut the gap to 18-10 before the break, Raguin misjudging Danny McGuire's kick to the wing and Hall pouncing to score.
Leeds made the perfect start to the second half in front of a record Dragons attendance of 9,880 as Burrow shot through a gap to score. Kevin Sinfield converted.
Hall made another telling error when knocking on after Catalans had failed to find touch from a penalty and Elima powered through to the posts for his second try, Bosc making it 24-16.
But Elima turned to villain in the 53rd minute when he lost the ball on halfway and from the scrum, Burrow found space to send McGuire over.
The Rhinos took the lead for the first time when Sinfield's inside ball freed Lauitiiti to score. Sinfield converted to make it 24-26.
With six minutes left, an innocuous kick on the last tackle from Julien Touxagas found Donald in space and he scampered 70 metres - evading three tacklers - for the try of the game.
Sinfield's third miss kept the match alive but his drop goal three minutes later sealed it and in the closing minute Hall atoned for his early errors with his second touchdown.