The result at Stade Lille Metropole leaves the visiting side just four points off the final Champions League place in fourth for the time being.
Lille had the better of the chances in the opening exchanges as Dimitri Payet's effort went wide on two separate occasions.
Civelli headed wide from a corner after the restart, before Aurelien Chedjou headed Payet's free kick just high for the home side.
Payet went close to breaking the deadlock midway through the second half when he shot on goal from outside the Nice penalty box, but was denied by a good save from goalkeeper David Ospina.
It was Nice who scored the opening goal of the evening and it came on 77 minutes, Civelli rising above others to head home Eric Bautheac's delivery from close range.