The home side took the lead after just five minutes when Dimitar Berbatov delicately lobbed the goalkeeper to make it 1-0.
Monaco continued to dominate the opening stages as Nice struggled to contain Claudio Ranieri's side.
Valere Germain hit the post with a strike on 16 minutes as Monaco pressed Nice into the own half.
The away side had their first real effort on 20 minutes through Valentin Eysseric, but the midfielder's shot was parried away by the goalkeeper.
Monaco kept hitting Nice on the counter-attack, putting David Ospina under constant pressure each time they moved forward at pace.
Nice had a shout for a penalty turned down by the referee, who ruled that Andrea Raggi didn't deliberately handball in the area.