The Ospreys' hopes of reaching the Heineken Cup quarter-finals suffered a huge blow after being beaten in France.
Irish wing Tommy Bowe gave the Welsh region an early lead when he crossed after just seven minutes.
But they were second-best after that as Clermont took a 12-7 half-time lead thanks to the boot of Brock James.
Julien Malzieu claimed the hosts' first try after Lee Byrne spilled the ball and Napolioni Nalaga wrapped up the win after intercepting James Hook's pass.
The defeat means the Ospreys must beat Leicester at the Liberty Stadium next weekend to have any hope of qualifying for the last eight.
It looked early on as they would not be left with that worst-case scenario after making a bright start.
Bowe could have scored inside the first minute but he could not stay on his feet as he collected Byrne's pass with a clear run ahead of him.
Luck then appeared to be on the Ospreys' side as Nalaga beat captain Ryan Jones to an excellent kick-and-chase ball from Anthony Floch but video replays showed he knocked the ball on before grounding it over the line.
After seven minutes, Bowe made amends from his earlier slip when once again Byrne was the supplier for him to go over in the corner to maintain his 100% record in the competition this season, with Dan Biggar converting.
They could have doubled their try count four minutes later when Ricky Januarie stole the ball from Nalaga in his own 22 but Seremaia Bai's superb saving tackle prevented the South African scrum-half from going over.
Tommy Bowe has scored in every Heineken Cup game this season
Clermont then started to pile on the pressure and James kicked three penalties and a drop-goal to give them a 12-7 lead at the break.
The second half started as the first half ended with Clermont in control, although James missed two penalties while Biggar missed a chance for the Ospreys.
But it was Malzieu who struck the killer blow shortly after the hour mark.
Byrne spilled James' high up and under and Aurelien Rougerie spotted the unmarked wing, who ran home. James missed the conversion, striking the post for a third time.
The Ospreys' misery was completed in the closing stages after Adam Jones was sin-binned for kicking the ball away.
James kicked his fourth penalty before Nalaga sealed victory for Clermont with three minutes to go after Hook's pass was intercepted.
Ospreys coach Sean Holley:
"We came out here with a lot of ambition and hope. We felt we had a real chance to win the game.
"A lot of credit goes to the opponents who were particularly outstanding.
"We've got to lick our wounds and pick ourselves up for next week. We saw a team today who I think will go a long way in this competition."
Clermont fly-half Brock James:
"We limited our errors and I think we put a lot of pressure on them in field possession.
"They got off to a flier... but it was important for us to get our hands on the ball and control the play a little bit.
"We knew they were going to come here and try to win and they put up a really good fight."
Clermont Auvergne: Floch, Nalaga, Rougerie, Baikeinuku, J. Malzieu, James, Parra, Domingo, Ledesma, Zirakashvili, Cudmore, Privat, Bonnaire, Lapandry, Vermeulen. Replacements: Russell for Floch (68), Canale for Baikeinuku (58), Debaty for Domingo (53), Scelzo for Zirakashvili (66), Audebert for Vermeulen (62). Not Used: Wepener, Pierre, Senio.
Ospreys: Byrne, Bowe, Parker, Bishop, Walker, Biggar, Januarie, James, Hibbard, A. Jones, A. Jones, J. Thomas, Collins, Holah, R. Jones. Replacements: Bennett for Hibbard (66), Tiatia for Collins (72). Not Used: Bevington, Griffiths, Gough, Nutbrown, Hook, Williams.
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