Elie Baup's side - who were undone by a 2-1 loss to Paris Saint-Germain in their last outing - were unable to find a response to Dario Cvitanich's strike at the Allianz Riviera.
With the match looking likely to head into half-time goalless, Nice broke the deadlock in the 40th minute when the Argentine forward sprung the offside trap before cleverly rounding Steve Mandana and slotting into the empty goal.
Marseille pushed for the equaliser but were unable to break down a stubborn Nice defence, with Colombian goalkeeper David Ospina on form.
It extends their winless run in Ligue 1 to seven matches, although Hubert Fournier's side had drawn their previous six league matches.
And Reims were denied a share of the spoils with two minutes remaining thanks to Martin Braithwaite's penalty.
