Monaco were frustrated by their 10-man French Riviera neighbours for 114 minutes until second-half substitute Berbatov came to the rescue by tapping home his first goal since arriving on loan from English Premier League Fulham last month.
Timothee Kolodziejczak of Nice was given his marching orders after slapping former Chelsea and Real Madrid defender Ricardo Carvalho just after halftime.
Mounir Obbadi and Lucas Ocampos went close for Monaco after the sending-off but stubborn Nice did well to take the tie into an extra half-hour.
Earlier, the visitors had a penalty claim waved away in the 39th minute when defender Andrea Raggi broke into the area and drew contact from Mathieu Bodmer.
Monaco also went close through Anthony Martial and Yannick Ferreira Carrasco before the interval.
Elsewhere, second-tier Auxerre lost 1-0 at home to Stade Rennes while Guingamp defeated fourth-tier Balagne Ile Rousse 2-0.
(Reporting by Mark Pangallo; editing by Tony Jimenez)
