Wales internationals Justin Tipuric and Eli Walker then clawed the GUINNESS PRO12 side back into it, only for David Strettle and Wesley Fofana to extend Clermont's lead out once again.
However, Ospreys' never-say-die attitude shone through as Sam Parry and Tom Habberfield, who finished off a wonderful team try at the death, crossed to clinch both the losing and try-bonus points in a 34-29 thriller.
And Biggar says fearlessness was to key to his side's superb second-half display.
"What happened is just a huge testament to the boys," said Biggar.
"It's the best club atmosphere in the world, an absolutely incredible stadium out here and the boys rose to the challenge.
"We came here and thought: 'Were we going to come away with a losing bonus point by kicking penalties and grinding a result out'? Probably not.
"We said we want to put a little bit of pressure on them and see if we could unearth a few cracks and really test them. To be fair to Clermont, they stood tall for the first half.
"At half-time, we said: 'Look, we have got to keep going, we aren't going to peg our way back by kicking penalties, we have got to chance our arm, if it comes off or we lose by 40 points we have given it a go'.
"We just managed to get the balance right between playing short and flying the ball wide."
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