Clermont captain Aurelien Rougerie got two tries before the break, with fellow wing Julien Malzieu also crossing.
Dafydd James scored for the Scarlets but they trailed 20-7 at half-time.
Julien Bonnaire got a fourth for the French, Scarlets replied through Dwayne Peel and Regan King, but three more Clermont tries sealed a French win.
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