The Rhinos rose to sixth in the Stobart Super League table thanks to a 10-try display, six coming in the final 20 minutes of the contest.
The hosts had hoped to make it back-to-back wins at home but were stunned by a late surge as Shaun Lunt (2), Rob Burrow, Carl Abeltt, Zak Hardaker and Kylie Leuluai also claimed tries with Kevin Sinfield successful with nine conversion attempts.
The Broncos managed a try in each half, courtesy of Chris Bailey - who had actually given them an early lead - and Kieran Dixon which were both converted by Michael Witt.
Catalan Dragons closed within a point of Warrington with a battling 34-10 win over the Wakefield Wildcats.
Damien Blanch (3), Steve Menzies (2) and Clint Greenshields scored the tries for the home side in the energy-sapping heat of Perpignan.
It enabled the Dragons to reclaim third place from St Helens and made sure they took full advantage of Friday defeats for leaders Wigan and second-placed Warrington.
Scott Dureau kicked five conversions who led 18-6 at the interval after Ali Lauitiiti claimed a try for Wakefield.
Kyle Trout gave them hope with a second early in the second-half, but the Dragons pulled away once more to end their challenge.