Tigers trailed 16-0 after 35 minutes at Stade Marcel-Michelin, where Clermont were unbeaten in their previous 74 games in all competitions.
Jordan Crane’s try and the boot of Owen Williams then saw former champions Leicester recover to within a score of a remarkable win.
However, Thomas Waldrom’s untimely yellow card proved costly as Clermont, last year’s beaten finalists, held out for a 22-16 win to book their place in the semi-finals for the third successive season.
Leicester director of rugby Cockerill said: “Gallant losers is not my thing. Yes we can take the positives from how we played, but we came with the attitude to win and we didn’t. Clermont are a very good side and you are always going to be up against it in France and especially here, where they played very well in the first half. Our plan was to try and keep the scores as close as possible in the first half and then go for it in the second half, but in the end they had just too much of a lead.”
France centre Wesley Fofana crossed on 22 minutes as Clermont surged 16-0 ahead in front of a raucous home crowd. But Leicester’s recovery began three minutes before half-time when fly-half Williams found Jamie Gibson in space with a cross-field kick and Blaine Scully handed Crane an easy finish.
Williams converted and then added three successive second-half penalties, while Morgan Parra and Brock James fired wide from distance, to slash the deficit to just three points at 19-16 on the hour.
Cockerill added: “We played well in parts and we have to take credit for the way we came back. We could have been obliterated because all the momentum was against us, they were leading and the crowd were singing. It was a tough situation.
“But fair play to our players, they took themselves to places they have never been in the Premiership or even in Europe at Ulster and Montpellier. To come back against a team of Clermont’s strength and come within a whisker of winning takes some doing. “Were Clermont lucky to win? Probably not. Were we unlucky to lose? Perhaps, but that’s rugby sometimes.”
Clermont have never lifted the Heineken Cup but hope they can go one better than last year’s agonising loss in the final to rivals Toulon in the Aviva Stadium.
Head coach Vern Cotter said: “It’s a major achievement to reach the semi-finals of such a great competition. The first-half performance was the best we have produced for a long time. But our experience from previous big games such as last year’s final was also crucial in the second half. It helped us both in attack and defence to raise our level when we really needed.”
Scorers for Clermont Auvergne: Tries: Fofana. Cons: Parra. Pens: Parra 5.
Scorers for Leicester: Tries: Crane. Cons: O Williams. Pens: O Williams 3.
CLERMONT AUVERGNE: Buttin, Nakaitaci, Rougerie, Fofana, Nalaga, James, Parra, Domingo, Kayser, Zirakashvili, Cudmore, Hines, Bonnaire, Chouly, Lee.
Replacements: Debaty for Domingo (50), Paulo for Kayser (52), Ric for Zirakashvili (69), Pierre for Hines (76), Bardy for Lee (71).
LEICESTER: Tait, Scully, Tuilagi, Allen, Goneva, O. Williams, B Youngs, Ayerza, T Youngs, Mulipola, Deacon, Slater, Gibson, Salvi, Crane.
Replacements: Flood for O Williams (76), Mele for B Youngs (76), Stankovich for Ayerza (76), Kitchener for Deacon (58), Waldrom for Crane (62).
Referee: Alain Rolland (Ireland).