Leicester were second best for most of the game and now need to win the return clash at Welford Road next weekend.
The Tigers, who had won their previous two pool games against Italian outfit Aironi and Ulster, paid a heavy price for their own indiscipline.
Hooker George Chuter and England centre Manu Tuilagi were both sin-binned in the 50th minute for dangerous play, leaving the Tigers with 13-men for 10 minutes.
Clermont made the most of their two-man advantage to record their 35th consecutive home victory in front of a partisan 18,000 crowd at the intimidating Stade Marcel Michelin.
Leicester scored the first try of the game totally against the run of play when Clermont's fly-half Brock James had his grubber kick charged down.
Toby Flood's fly-hack deep into Clermont's territory created confusion in the home side's defence and saw scrum-half Ben Youngs run in at the corner to give the visitors an improbable 5-3 lead.
Scrum-half Morgan Parra calmed the home side's nerves when he slotted over his second and third penalties to reclaim the lead midway through the first half.
The home side then stretched their lead when burly wing Julien Malzieu barged his way through some weak tackling and crashed over from short range to make it 16-5 at half-time.
After the break the Tigers were under extreme pressure and lost two players to yellow cards, hooker Chuter and centre Tuilagi, for dangerous play.
The pair were punished for high tackles on Para and Vincent Debaty respectively in the same attacking move and the Tigers never recovered.
Clermont turned the screw when impressive centre Wesley Fofana crossed for the French side's second try midway through the second half.
Fofana rubbed salt into the wounds with his second try before the visitors were awarded a consolation penalty try after Clermont collapsed a scrum in the shadow of their own posts.
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