Having led Ligue 1 in early January, Marcelo Bielsa's side have dropped out of title contention and their latest loss leaves them battling for a Europa League spot.
Defender Cedric Yambere scored his first Bordeaux goal just after the hour to seal all three points for the hosts, who sit in sixth and are now just three points behind Marseille.
After their Coupe de France semi-final exit to Paris Saint-Germain last week, Saint-Etienne bounced straight back in the league with a 1-0 win over Nantes.
Franck Tabanou grabbed the decisive goal after 18 minutes to keep Christophe Galtier's side in top-four contention.
Toulouse boosted their hopes of survival against Montpellier, Oscar Trejo netting on the stroke of half-time to secure a 1-0 victory.
Bottom club Lens kept their faint hopes of staying up alive after a goalless stalemate with Lorient, the draw moving them to within nine points of safety with six games remaining.
Meanwhile, Lille struck late on through Sofiane Boufal to beat Evian 1-0, Rennes triumphed over 10-man Guingamp by the same scoreline and Nice moved into the top half as they also won 1-0 against Reims.
