Their early dominance continued to pay off as Irish winger Tommy Bowe claimed the first try eight minutes later after a neat chip by Nikki Walker.
Biggar was unable to convert and it allowed Clermont back into the game. A kick-and-drive move ended with a drop goal for fly-half Brock James.
But the visitors lost lock Thibaut Privat to a yellow card for tripping Shane Williams and the Ospreys took advantage of the extra man.
They got their second try from a scrum after James spilled in front of his own posts 2m out. Mike Phillips was denied at the line, but captain Ryan Jones completed the move.
That gave the Ospreys a 15-3 lead, but also found themselves down to 14 when Jerry Collins needlessly shoulder charged James off the ball.
They were to suffer a further blow when Wales scrum-half Phillips was forced off seven minutes before the break with an ankle injury.
It was the end of the game for Mike Phillips who went off with a foot injury
It did not seem to affect them though as they claimed a superb third try right at the end of the first half.
Bowe raced on to Biggar's chip and he off-loaded to Hook who found full-back Barry Davies, in for the injured Lee Byrne, in space to burst through. Biggar converted for a 22-3 half-time lead.
Clermont needed to get themselves back into the game quickly after the restart and they did just that with Benoit Baby touching down Julien Bonnaire's chip in the corner for a converted try.
It immediately put the Ospreys under pressure, who were hoping to avoid a repeat of last week's fight-back by Leicester.
But just 10 minutes into the second half, Clermont got their second through replacement Thomas Domingo, which James converted to cut the gap to 22-17.
And moments later, the visitors completed a stunning comeback, albeit in controversial circumstances.
Domingo broke through and offloaded to Mario Ladesma, but he appeared to send the ball forward to Alexandre Lapandry before the hooker went over. James made no mistake to give the French side the lead for the first time.
The Ospreys managed to reorganise their suffering scrum which had needed bolstering, and won a penalty in which Biggar kicked to restore the lead.
Clermont thought they got themselves back in front in the corner but Napolini Vonowale Nalaga's foot was in touch as he went over.
It created a pulsating climax and the Ospreys survived late on when James sent a penalty against the post from 40m.
Ospreys coach Sean Holley: "We went out with real intent and fortunately a lot of it paid off."
"The fact that Leicester and Clermont have come back into the game is just a measure of how the game ebbs and flows.
"Their third was a little bit questionable but it put us under a lot of pressure and we had some difficult moments at the scrum at times.
"To get six points out of our first two games is a huge bonus."
Clermont coach Vern Cotter: "I think we didn't get our hands on the ball enough in the first half.
"I thought the Ospreys started well and scored points and we couldn't produce and maintain pressure.
"In the second half we got our hands on the ball more and were able to express ourselves more.
"I was very happy with the character we showed in the second half. We had a fourth try disallowed and a penalty hit a post so we were in with a good chance of winning it at the end."
Ospreys: B. Davies, Walker, Bowe, Hook, Williams, Biggar, Phillips, D. Jones, Bennett, Griffiths, A. Jones, Thomas, Collins, Holah, R. Jones, Nutbrown. Replacements: Nutbrown for Phillips (33), James for D. Jones (55), Hibbard for Bennett (36), Mitchell for Griffiths (55), Gough for Thomas (57), Tiatia for Holah (59), Bishop for Nutbrown (73). Not Used: Parker.
Clermont Auvergne: Rougerie, Baby, Canale, Baikeinuku, Nalaga, James, Senio, Faure, Ledesma, Scelzo, Cudmore, Privat, Bonnaire, Lapandry, Vermeulen. Replacements: Russell for James (75), Domingo for Faure (47), Debaty for Scelzo (56), Bardy for Lapandry (66). Not Used: Wepener, White, Parra, Williams.
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