Amazingly, Biggar has claimed all Ospreys' 55 points in the competition this year.
For Castres, who failed to get a losing bonus point, scrum-half Rory Kockott kicked all the points with four penalties.
There was little at stake in this game for the Ospreys in terms of Heineken qualification after they had lost their opening three matches to Leinster, Northampton and Castres.
The Ospreys were forced into a late change with lock Ian Evans pulling out at the last minute after his wife was admitted to hospital as a precaution a week after giving birth. He was replaced by Tyler Ardron from the bench.
Skipper Alun Wyn Jones was rested with Ryan Jones taking over the captaincy, but they did have hooker Richard Hibbard back.
Sam Davies, who help guide Wales Under-20s to the World Cup final last summer, made his Heineken Cup debut at full-back in place of Richard Fussell.
The first half turned into a kicking contest as Biggar and Kockott swapped four penalties each. All Castres' four transgressions came in and around the 22, with captain Remy Tales paying the price after the fourth penalty with a yellow card on 28 minutes for a professional foul.
Kockott's fourth penalty drew the scores level at 12-12 in the final minute of the first half.
The Ospreys might have gone into the break on level terms but they struggled up front, especially in the scrum, and also lost a crucial line-out in the French side's 22.
Castres, despite losing Richie Gray with a knee injury on 25 minutes, did look the more dangerous although wing Jeff Hassler had an excellent run into the 22 which did earn the Ospreys one of their four penalties.
In the second half the Ospreys gradually got a modicum of control and turned around their problems in the scrum. All three of Biggar's second-half penalties came from Castres' infringements at the set-piece.
Ospreys were on the front foot when Biggar put his side into the lead after only three minutes of the second period.
And he added two more penalties on 61 and 70 minutes to take Castres out of losing bonus point range.
In the dying minutes Castres tried their best to reduce the deficit, but ran out of time.
- Sports & Recreation
- Rory Kockott