Takudzwa Ngwenya scored an incredible 80-metre try while Damien Traille booted three drop goals to break the Welsh region’s hearts in San Sebastian.
The Ospreys outscored the French giants three tries to two through skipper Ryan Jones, Lee Byrne and Nikki Walker.
And it is the third successive time the star-studded side have exited Europe’s premier competition at the quarter-final stage.
Ospreys director of coaching Scott Johnson was fit enough to travel to northern Spain despite his health scare last weekend. But the Australian would have been disappointed with his side’s start when Biarritz centre Damien Traille scored a 29-second drop goal.
The Ospreys rallied and began to show their attacking threat with a series of near-misses inside the opening ten minutes.
Byrne, James Hook and Andrew Bishop cut through the French defence with ease but failed to produce a clinical finish.
The British Lions full-back sliced open Biarritz with a thrilling run but Shane Williams was dumped into touch and Bishop failed to take full advantage when Iain Balshaw fumbled.
The difference in finishing was exposed in the 12th minute when the home side took the lead in devastating fashion.
Ngwenya picked up the ball inside his own 22 and beat off the tackles of Mike Phillips and Williams for a superb 80-metre touchdown.
Dimitri Yachvili added the extras for a 10-point lead but the Welsh region responded at the end of a frustrating first quarter through a rare try from Jones.
The skipper started and finished a flowing move which involved neat work from strike-runners Byrne, Williams and Hook before the No8 dummied Ngwenya to crash over for a converted try.
Biarritz moved further ahead through Yachvili’s long range penalty and Traille’s second drop goal but the Ospreys were let off the hook when Yachvili missed the chance to further increase their advantage with a 28th minute penalty.
Then the fit-again Alun Wyn Jones and Tommy Bowe teamed up to produce a try-scoring tackle on flanker Wenceslas Lauret.
And their last-ditch efforts proved crucial as the Ospreys roared back before the interval with an excellent team try five minutes before the break.
Both forwards and backs combined in a neat move before Marty Holah sent the full-back Byrne home in the right corner.
Biggar missed with the touchline conversion but the confident fly-half completed a thrilling by booting Johnson’s men to within one point with a deft drop goal.
The young pivot then fired the Ospreys into the lead for the first time but Biggar turned from hero to zero within moments when he conceded a soft penalty for tackling Ngwenya without the ball and then failed to kick the ball ten metres from the re-start.
Biarritz did not waste any time in taking full advantage and Balshaw pounced on Ngwenya punt ahead and fended off Bishop to touchdown.
Referee George Clancy was forced to check with television match official Keane to see if the England full-back had grounded the ball safely but the Irishman awarded the try.
Yachvili converted and Traille sealed a hat-trick of drop goals either side of Biggar’s second penalty to keep Biarritz 29-21 in front.
The Ospreys threw on Nikki Walker for Bishop in an attacking move but Biggar failed to reduce the deficit when he saw his penalty and drop goal attempt fly wide of the posts.
The visitors maintained the pressure and were rewarded with a third try through replacement Walker.
Scrum-half Phillips found a gap in the Biarritz midfield and his inside pass allowed the Scottish winger to score under the posts.
Biggar’s conversion brought the scores to within one point and set up a grandstand finish.
