The French side now await the winner of the second semi-final between Saracens and Toulon, which takes place at Twickenham tomorrow.
It was the visitors who went ahead as early as the seventh minute, when Ronan O'Gara kicked a penalty right between the sticks.
However, that lead was short-lived, as Napolioni Nalaga danced through to touch down, before Brock James converted for a 7-3 advantage.
The Aussie fly-half then kicked the home side further ahead with a penalty just moments later.
Dennis Hurley touched down after some great work by O'Gara and the fly-half then converted to get his team right back in the match.
The momentum swung in favour of the Red Army, who threw everything that they had at Clermont in a brave effort to reverse the deficit.
However, they could not pull off a miraculous comeback by taking advantage of the nervy defending and will return home disappointed.