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Francis Gillot's men opened the scoring five minutes shy of the interval when Traore smashed home from close range to break a two-and-a-half-year goal drought.
The hosts doubled their advantage six minutes into the second half, with Henrique sweeping home from close range after Nicolas Maurice-Belay's effort had rebounded off the post.
Gregory Sertic came within inches of adding to his team's two-goal lead by curling a free kick narrowly past the post on the hour mark.
Saint-Etienne could have set up a nervy final few minutes for Bordeaux as Brandao saw his header expertly saved by Cedric Carrasso in the closing stages of normal time, but the home team held out for a clean sheet.