Also Friday, Toulouse beat Reims 2-1 to provisionally climb to seventh place.
Oscar Trejo gave Toulouse the lead with a header in the 42nd before Reims midfielder Prince Oniangue equalized with a curling shot from the edge of the box in the 56th.
Toulouse forward Martin Braithwaite converted a penalty in the 86th after Reims midfielder Antoine Devaux received a red card for fouling Clement Chantome.
Mid-table Reims stretched its winless streak to seven games.
On Saturday, it's: Paris Saint-Germain vs. Bastia; Ajaccio vs. Nantes; Evian vs. Guingamp; Montpellier vs. Lille; and Rennes vs. Valenciennes.
On Sunday, Monaco visits Sochaux, Lyon hosts Bordeaux and Saint-Etienne faces Lorient.
Marseille threatened early with crosses from the left flank. Mathieu Valbuena found Andre Ayew in the sixth minute but the Ghana winger headed over the bar. Then Florian Thauvin met a cross from leftback Jeremy Morel in the 15th but Nice goalkeeper David Ospina turned his 16-yard volley around the post.
The Marseille defense was forced to reorganize itself in the eighth minute when Souleymane Diawara replaced Brazilian centerback Lucas Mendes, who picked up an ankle injury.
Nice capitalized on that substitution to find space inside the Marseille area. An unmarked Nemanja Pejcinovic knocked the ball down for Cvitanich in the 10th instead of attempting to score with a header and Marseille defender Nicolas Nkoulou was able to clear the danger.
Nice midfielder Mahamane Traore also had plenty of space in the 22nd but missed the target from 12 yards after a cross from Kolodziejczak.
Nice grew more confident and came close to scoring in the 31st. Cvitanich controlled a cross from Kolodziejczak to feed winger Jeremy Pied, whose powerful half-volley crashed against the bar.
Nice finally broke the deadlock in the 40th. Cvitanich beat the offside trap to round goalkeeper Steve Mandanda for his sixth goal this season.
Marseille's best chances in the first half came from set pieces. A corner from Thauvin eluded the Nice defense in the 34th, but Diawara missed a tap-in from five yards. On the stroke of halftime, Traore protected the hosts' lead by clearing off the line Andre-Pierre Gignac's header from a corner.
Marseille pushed for an equalizer in the second half but Ospina denied Giannelli Imbula in the 46th, parried a powerful strike from Thauvin in the 56th before stopping a header from Gignac in the 79th.
Marseille also lacked the finishing touch twice. In the 67th, Valbuena played Ayew in but the winger's close-range effort hit the post. Ayew then connected with a free kick from Valbuena in the 76th but sent his volley wide.
In stoppage time, Ospina preserved his team's victory by saving a header from Ayew.
"We could have been more accurate at times but Ospina had a great game," Marseille coach Elie Baup said. "I don't blame my players. They tried everything in the second half to come back."
Ospina helped Colombia win 2-1 Tuesday in Paraguay to qualify for the 2014 World Cup.
