Thierry Henry's side was held 1-1 at home by Patrick Vieira's Nice despite the visitors having Ihsan Sacko sent off at the end of the first half.
Benoit Badiashile Mukinayi headed a 50th-minute equaliser to cancel out the opening goal scored by Allan Saint-Maximin at Stade Louis II on Wednesday.
Monaco could not force a winner to boost its survival hopes and even had to survive a late penalty scare against the 10-man visitor.
With 13 minutes to go, Saint-Maximin was denied by Monaco goalkeeper Diego Benaglio after a VAR review adjudged Youcef Atal to have been fouled in the box.
Henry's Monaco remains 19th in the table, which ensured it is on its worst home run in a single Ligue 1 season.