Clermont trailed 16-3 after 25 minutes when Leigh Halfpenny followed up three early penalties by converting a Charles Ollivon try.
However, the visitors came roaring back and moved ahead in the second half as Patricio Fernandez crossed twice in the space of two minutes.
Toulon were ultimately grateful for a 73rd-minute try from Mamuka Gorgodze, with Pierre Bernard adding the extras to ensure the lead changed hands once again.
Even then the drama was not over, Camille Lopez missing the target with a late drop-goal attempt that would have earned Clermont victory.
Prior to the match, Toulon president Mourad Boudjellal confirmed former Bath chief Mike Ford will succeed Steve Meehan as the club's backs coach.
After beginning the season with successive defeats, Montpellier have now claimed four wins in a row following a 42-13 thrashing of Brive.
Nemani Nadolo claimed a hat-trick for Montpellier, who ruthlessly punished yellow cards for Sisa Koyamaibole and Teddy Iribaren.