The visitors' veteran forward Jimmy Briand equalised from the spot in the second half after Pierre Lees-Melou scored for Bordeux within the opening half an hour.
Paulo Sousa kept faith in the front-three which were clinical in last weekend's 2-0 victory over Nantes as in-form South Korea international Hwang Ui-jo started on the wing.
Patrick Vieira gave attacker Adam Ounas his first appearance since late-September after he recovered from a knee injury.
Lees-Melou, who grew up near Bordeaux, picked up his second yellow card in as many games after less than eight minutes.
He made up for his earlier error by opening the scoring for the hosts after 27 minutes to claim his first goal of the campaign.
Wing-back Youcef Atal found Lees-Melou with a whipping cross and the midfielder headed home.
Denmark's Kasper Dolberg could have double the home side's advantage two minutes later but Benoit Costil blocked the former Ajax forward's rushed shot with his feet.
Aurelien Tchouameni had a chance to equalise for Bordeaux seven minutes from half-time but Walter Benitez tipped his curling effort over the crossbar and Nice led at the break.
Less than two minutes into the final 45 Bordeaux were handed a golden chance from the penalty spot after Stanley Nsoki brought down winger Francois Kamano.
Former France striker Briand stepped up to beat Benitez for the 34-year-old's first goal in seven games.
Atal continued to threaten in wide spaces down the right wing for Vieira's side as Sousa replaced Kamano with English youngster Josh Maja to try and break the deadlock but neither side could find a winner.
On Saturday, Paris Saint-Germain travel to Brest with striker Edinson Cavani set to make his first start for the champions since August.
Rudi Garcia takes Lyon to Marseille on Sunday, a club he managed for three years, without their top goal scorer Memphis Depay.