It was the away side who came closest to opening the scoring after four minutes when Ludovic Obraniak advanced on goal, but his effort drifted just high of the post.
Benoit Tremoulinas was next to go close for the away side after 23 minutes, but his effort was pushed away by Elana.
The away side eventually found the goal that their play deserved after 35 minutes when Gouffran found the bottom corner following a set piece.
Bordeaux moved into a 2-0 lead early in the second period when Jaroslav Plasil headed a Gouffran cross into the top corner of the net.
Obraniak missed a golden chance to all but finish the home side off after 61 minutes, but his header just missed the target.
Brest missed a golden chance to cut the score to 2-1 after 76 minutes when Bruno Grougi missed a penalty after Obraniak had felled him in the box.
Bordeaux played out the final few moments with relative ease, but they remain in ninth place in Ligue 1.