Bobo's double proved decisive
Biarritz (17) 27
Tries: Traille, Noirot, Bidabe, Bobo 2
Border Reivers: (3) 17
Tries: Palepoi, MacRae
Cons: MacRae 2
Borders came within three minutes of snatching a shock draw before two late tries gave Biarritz victory and denied the visitors a bonus point consolation.
The hosts led 17-3 at half-time after tries from Damien Traille, Benjamin Noirot and Philippe Bidabe.
But the Borders, who had only managed a Calum MacRae penalty, hit back to level when lock Opeta Palepoi and then MacRae crossed, MacRae converting both scores.
But Fijian wing Sereli Bobo broke Borders hearts with two late tries.
Replacement Dimitri Yachvili created both, first catching the visitors' defence on the hop with a cross kick to the unmarked Bobo - when the Borders were expecting a shot at goal.
Then a Yachvili dart created the time and space for Bobo to beat the exhausted Reivers markers for pace in the dying seconds.
Biarritz had the bulk of possession and territory in the early stages but a series of kicking and handling errors prompted howls of derision from their own fans.
MacRae missed an opportunity to put Borders ahead with a penalty attempt that drifted wide.
Instead the hosts went in front when after a powerful Santiago Delappe surge, stand-off Traille brushed off Gregor Townsend's weak tackle to score, Julien Dupuy converting.
Biarritz scored their second try from another penalty kicked to touch, driving hooker Noirot over from the line-out.
The third arrived after five scrums in quick succession near the Borders line, number eight Thomas Lievremont picking up to send Bidabe over.
Reivers gave themselves a glimmer of hope in the dying seconds of the half as MacRae landed a penalty, and they stunned the hosts with two tries in the third quarter.
After Serge Betsen and Andy Miller were yellow-carded for fighting, and replacement Biarritz hooker Denis Avril followed them to the sin-bin, the Scots seized their opportunity.
Palepoi was driven over from the penalty line-out, before MacRae side-stepped his way over from 35m, converting both to make it 17-17.
But Yachvili's inspired contribution ended the Borders' hopes, and kept last year's finalists on track for a home quarter-final.
Biarritz: Dambielle; Gobelet, Bosch, Masi, Bobo; Traille, Dupuy; Balan, Noirot, Lecouls; Dellape, Couzinet, Betsen, Dridi, Lievremont.
Replacements: August, Avril, Malonga, Harinordoquy, Yachvili, Peyrelongue, Bidabe.
Border Reivers: MacRae (capt); Law, De Luca, Rennie, Daniel; Townsend, Chrystie; Kerr, Scott, Kalman, Grimes, Palepoi, Gray, Miller, Brown.
Replacements: Ford, Douglas, Dalziel, Newlands, McKerchar, Jones, Flockhart.