The minnows from Normandy pulled off the upset in a dramatic finale. They entered second half stoppage time trailing to Younousse Sankhare’s first-half opener for the visitors.
But Sullivan Martinet levelled with seconds left on the clock to send the tie into extra-time.
The first 15 minutes of that extra half an hour proved disastrous for Jocelyn Gourvennec’s men. After losing Youssouf Sabaly for two bookable offences just before Martinet’s equaliser, Thomas Carrique received his second yellow card for a foul on Ladislas Douniama in the penalty area.
Douniama, who had missed a spot kick during regulation time, dispatched it past Bordeaux goalkeeper Benoit Costil to give his side a 2-1 lead..
There was no way back for Bordeaux when Jaroslav Plasil was given a straight red to reduce his side to eight with 10 minutes remaining.
Gourvennec’s position will be under intense scrutiny ahead of the return to Ligue 1 action on 13 January when his team travels to Troyes. Since their 6-2 defeat at the Parc des Princes against Paris Saint-Germain in September, Bordeaux have won only one of their last 10 games in all competitions. They have slipped to 15th just one point above the relegation zone.