Willy Sagnol's side had been the early pacesetters after winning their first three games, but a draw and defeat in their last two tempered their progress.
However, Diego Rolan and Wahbi Khazri struck in either half to get the Garonne club back to form.
Yeltsin Tejeda, a star of Costa Rica's World Cup campaign, grabbed his first goal in French football late on but it was not enough for Pascal Dupraz's strugglers, who have conceded 15 goals in their first six games this term and are rooted to the foot of the division.
Evian had started strongly at the Stade Chaban-Delmas, but Bordeaux eventually began to get on top and Rolan should have done better when unmarked 15 minutes in but his flicked header was off-target.
The Uruguayan atoned for that miss just two minutes later though, flicking in at the near post of Nicolas Maurice-Belay's whipped cross.
Bordeaux left-back Diego Contento was forced off midway through the opening half, after a stray elbow caused him a cut above the eye.
Evian took advantage of the former Bayern Munich man's absence down the right, Fabien Camus delivering well, but Nicolas Benezet made a mess of his chance from seven yards.
Just 11 minutes after the interval, Bordeaux had breathing space as Khazri picked up the ball in the left channel before stepping inside and curling a sumptuous effort into Jesper Hansen's top-left corner.
Tejeda's deflected effort gave the visitors brief hope, but it was not enough and they look to have much work ahead to salvage their season.