Yachvili is the fifth highest points scorer in Heineken Cup history
Bath were undone by the boot of Dimitri Yachvili as a clinical Biarritz emerged victorious at the Recreation Ground.
The home side led 8-0 at the break thanks to an Olly Barkley penalty and a Michael Claassens try.
Three Yachvili penalties edged the French side in front before Barkley sent over another three-pointer.
But when David Wilson was sin-binned late on, Yachvili landed the resulting kick before Bath inexplicably declined a drop-goal chance under the posts.
Bath did not appear to be missing the services of England captain Lewis Moody, who was ruled out with an eye injury, when they stormed into an eight-point lead over last season's runners-up after seven minutes.
Barkley recovered from an early miss to break the deadlock from the tee, and, three minutes later, the fly-half's neat offload was worked via Nick Abendanon and Matt Carraro to South African Claassens, who crossed with ease.
Barkley could not add the extras, and Bath survived a scare when Michael Bond rampaged up field, but a superbly executed Abendanon tackle on Takudzwa Ngwenya closed the door on the French.
However, Bath's advantage had been cut to a meagre two points within 11 minutes of the restart as French scrum-half Yachvili coolly slotted over two penalties.
In the 52nd minute, the Premiership side required the intervention of the Television Match Official, who decided that number eight Raphael Lakafia had made two movements as he stretched for the line following a prolonged period of Biarritz pressure.
Yachvili and Barkley exchanged penalties before prop Wilson saw yellow for hands in the ruck.
Former Gloucester man Yachvili's match-winning kick a few metres inside his team's half just about had the legs but despite being a man down Bath continued to put the phases together.
Their tactical naivety cost them with five minutes to go, though, and they emerged only with a losing bonus point.
With a good platform in front of the posts, Claassens elected to move the ball wide with Barkley perfectly placed behind him to try for a drop-goal, and they were unable to engineer another chance.
Bath coach Steve Meehan: "It's actually quite difficult to put into words. We played a very good first half - we got all the decisions right - but a few technical penalties went against us in the second half.
"And then with four minutes to go, we had an opportunity to win the match, but we didn't take it. It was a better performance than last week [against Gloucester], yet to come up short like that is very frustrating.
"We've got to grow up and take the points. It's straightforward, it is about being ruthless, getting the job done and getting it through the sticks. Simple.
"Drop-goals and penalties win games. Biarritz kick the penalties and keep the scoreboard ticking over. We had the opportunity and didn't take it."
Replacements: Flatman for Catt (53), Batty for Dixon (75), C. Fernandez Lobbe for Grewcock (63). Not Used: Bell, Skirving, McMillan, Biggs, Cuthbert.
Sin Bin: Wilson (71).
Biarritz: Haylett-Petty, Ngwenya, Gimenez, Bond, Balshaw, Peyrelongue, Yachvili, Coetzee, Terrain, Johnstone, Carizza, E. Lund, M. Lund, Harinordoquy, Lakafia.
Replacements: Mignardi for Bond (67), Traille for Peyrelongue (61), August for Terrain (51), Marconnet for Johnstone (51), Taele-Pvihi for Carizza (67), Guyot for Lakafia (73). Not Used: Barozzi, Bolakoro.
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