Lyon stay on 48 points from 23 games, 11 behind runaway leaders Paris St Germain and level on points with third-placed Olympique de Marseille who were held 2-2 at home by AS Monaco, who are in fourth spot a point behind.
Bordeaux, who under previous coach Jocelyn Gourvennec had lost their three previous home games before he was sacked, prevailed through a Nicolas de Preville goal and penalties by Malcom and Gaetan Laborde, all in the first half.
Lyon, who beat Monaco and PSG in their last two games, reduced the arrears before the break through Marcelo.
Bordeaux, who finished with 10 men when Otavio was sent off for a second yellow card, jumped to ninth on 29.
Former Uruguay international Poyet was appointed last week after Gourvennec was ousted following a string of poor results.
Former Chelsea midfielder Poyet left his position as Shanghai Shenhua coach last September.
Bordeaux's De Preville opened the scoring in the 22nd minute after a counter attack before Malcom doubled the tally five minutes later, converting a penalty after being brought down.
Marcelo pulled one back in the 44th when he headed home from Nabi Fekir's corner, only for Laborde to restore Bordeaux's two-goal advantage with another penalty just before the break.
At the Velodrome, Marseille were caught cold when Monaco's Keita Balde opened the scoring in the fourth minute, coolly slotting the ball past goalkeeper Steve Mandanda.
Three minutes later, Adil Rami equalised with a header.
Valere Germain put the hosts ahead two minutes into the second half with a diving header from Dimitri Payet's free kick.
But Fabinho earned a point for Monaco when he fired home from the edge of the box in the 51st minute.
(Reporting by Julien Pretot; Editing by Clare Fallon and Ken Ferris)
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