The Italian striker then claimed his second of the game on 68 minutes, latching on to a Jean Michael Seri pass before seeing his shot beat Danijel Subasic with the aid of a deflection off Andrea Raggi.
Nice looked like they were on the way to three points, but in injury time, Stevan Jovetic's shot was saved by Walter Benitez before Radamel Falcao turned in the rebound for his 16th league goal of the campaign.
The result keeps third-placed Monaco winless in 2018 and leaves them 10 points behind league leaders Paris Saint-Germain, while Nice remain sixth in the standings.
MONACO (4-2-3-1): Subasic; Sidibe, Glik, Raggi, Jorge; N'Doram (Ghezzal 81'), Moutinho; Keita (Lopes 60'), Lemar, Diakhaby (Jovetic 76'); Falcao
NICE (4-3-2-1): Benitez; Burner, Marlon, Dante, Le Marchand (Sarr 60'); Lees-Melou, Seri, Cyprien; Saint-Maximin, Plea; Balotelli (Srarfim 90+3')
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