Former England wing Iain Balshaw scored two tries for Biarritz, but his opposite number Tommy Bowe's second-half touchdown - and 23 points from Biggar - ensured a winning start in Pool Five for the Welsh regional side.
Biggar landed seven penalties and a conversion with his eight kicks at goal, finishing just three points short of equalliing Gavin Henson's Ospreys Heineken Cup record of 26 against Harlequins seven seasons ago.
Biggar's unswerving accuracy was responsible for the Ospreys extending their unbeaten European home record to 17 games since they lost at home to Leicester in December 2005.
At times they made hard work of it, struggling for fluency in their attacking game, but with Biarritz regularly conceding penalties Biggar was in no mood to let them off the hook.
He kept the scoreboard ticking over and allowed the Ospreys an early opportunity to take charge of their group.
Having been Heineken quarter-finalists in three of the last four seasons, the Ospreys are determined to go at least one step further this time around.
And victory over Italian hosts Treviso next weekend would send them into next month's double-header against Aviva Premiership champions Saracens in confident mood.
The Ospreys paraded four of Wales' World Cup squad as they faced their first Heineken Cup game without departed trio Mike Phillips, James Hook and Lee Byrne, who all sealed lucrative contracts in French club rugby earlier this year.
Biarritz, meanwhile, arrived in Wales needing to put dismal domestic form behind them that had seen go bottom of the Top 14 French championship without their World Cup players.
And the visitors showed they meant business from the start, dominating early possession and territory as centre Damien Traille dropped a fifth-minute goal either side of centre Marcelo Bosch and fly-half Julien Peyrelongue missing penalties.
It took the Ospreys almost 15 minutes to establish any sort of threatening position, but a long-range Biggar penalty hauled them level by the end of an error-strewn opening quarter.
A second Biggar penalty edged the Ospreys 6-3 ahead at half-time, but it was a dire opening period despite the best efforts of Biarritz skipper Imanol Harinordoquy to lift the contest from its below-par status.
Biggar completed his penalty hat-trick just two minutes into the second half, and when Biarritz infringed again shortly afterwards, he found his range from a testing angle 40 metres out.
It was an impressive effort by Biggar, who has been out of favour with Wales since last year's home draw against Fiji, slipping down coach Warren Gatland's fly-half pecking order.
A Bosch penalty brought Biarritz back to within losing bonus point range, but the French side's repeated indiscipline gave Biggar another penalty opportunity that he gratefully accepted for a 15-6 advantage after 50 minutes.
English referee Andrew Small readily punished both sides' technical deficiencies, and a second Bosch penalty in front of the Ospreys posts cut Biarritz's deficit to six points again.
But Biggar continued to prove a priceless points machine for the Ospreys, slotting his sixth successful penalty before Balshaw scampered over in the corner to give Biarritz hope.
That was the cue, though, for a concerted Ospreys response, and when Bosch failed to deal with a steepling kick just outside his 22, Ospreys scrum-half Kahn Fotuali'i reacted quickest and sent Bowe across.
Biggar inevitably converted and added another penalty, yet the Ospreys were quickly brought back down to earth when centre Andrew Bishop's pass was intercepted by Biarritz substitute Benoit Baby, who delivered a scoring ball to Balshaw.
Bosch's conversion ensured a nervous finale for the Ospreys, who should have been home and hosed at 28-14 clear, and they survived a disalllowed Balshaw try with just seven minutes left.
And that proved to be Biarritz's final opportunity of the game, although they had the consolation of leaving west Wales with a losing bonus, which could prove crucial in the final Pool Five shake-up.
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