Morgan Parra kicked two penalties from 12-7 down to seal the comeback win.
Naipolioni Nalaga opened the scoring on three minutes with a try in the corner following a break by Wesley Fofana.
France scrum-half Parra converted but the Parisians' full-back Benjamin Dambielle kicked three penalties to put them ahead.
Fabrice Estebanez sent a drop-goal over from long range on 56 minutes to strike fear into the home fans - but back came Clermont to keep their tremendous run going.
They now stand fourth in the table.