The Ospreys had already beaten Stade Francais, who they face next Friday night, in Swansea but the win was not enough to claim a quarter-final place.
However coach Lyn Jones will still hope this win can turn his team's season around following a dismal run of results and the loss off Lions trio Gavin Henson, Ryan Jones and Brent Cockbain for the near future.
Shane Williams was another injury concern and was left out of this game but the Ospreys other wingers - Terblanche and Mustoe - proved enough to beat a Clermont-Auvergne team badly missing the assuredness of crocked Stephen Jones.
Mustoe pounced when Clermont dropped the ball trying desperately to run the ball out of defence in the 35th minute. The in-form youngster hacked ahead before picking up for an easy finish under the posts.
Clermont finally hit back through centre Pierre-Manuel Garcia before bursts from Duncan Jones and Adrian Cashmore ended with Terblanche squeezing over in the corner. Cashmore went on to add 16 points with the boot.
Both teams were reduced to 14 minutes during a scrappy second half. Ospreys outside-half Shaun Connor was first to the sin bin following a scuffle before Gonzalo Longo followed seven minutes later.
Sam Broomhall crashed over on the hour mark but the Ospreys defence held firm before Cashmore rounded off a solid solo and team display with a late penalty.
Ospreys: Tries R Mustoe, S Terblanche. Cons - A Cashmore (2). Pens A Cashmore (3).
Clermont-Auvergne: Tries P-M Garcia, S Broomhall. Con C McMullen.