In a game that was played behind closed doors pending investigations into the incidents that caused their home game against Marseille to be abandoned last Sunday, Nice prevailed through an early goal by Justin Kluivert, a double by Amine Gouiri and Khephren Thuram's late strike.
Kluivert opened the scoring after just seven minutes when he volleyed home from Hicham Boudaoui's cross to give Christophe Galtier's side a perfect start.
He also assisted in the second goal, playing Gouiri through for the striker to finish the job with a left-footed effort on 33 minutes.
Gouiri made it 3-0 three minutes before the interval as he converted a penalty after being brought down by centre back Laurent Koscielny.
Thuram, the former France international Lilian's second son and brother of France striker Marcus, added another with five minutes left from inside the area shortly after coming on as a substitute.
Bordeaux, who have yet to win a game this season, are 17th with two points from four outings.
Later on Saturday, Marseille will be out to snatch their second victory of the season when they host Saint-Etienne.