Turkey internationals Yilmaz and Celik put Christophe Galtier's team on 76 points with three games left after champions Paris St Germain briefly topped the standings with a 2-1 home win against RC Lens earlier on Saturday.
Lille lead PSG by one point and third-placed AS Monaco, who host fourth-placed Olympique Lyonnais on Sunday, by five.
Nice, who had Jordan Lotomba sent off three minutes into the second half, are ninth on 46 points.
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Lille got off to a fine start with Yilmaz breaking the deadlock in the 13th minute, firing home a low shot from 18 metres after collecting a pass from Jonathan David.
Nice, who failed to muster a single attempt throughout the game, went down to 10 men when Lotomba picked up a second yellow card for a rough challenge on Jonathan Bamba and with him went their hopes of a comeback.
Celik added a second for Lille in the 56th minute with a superb volley from outside the area before another Yilmaz strike crashed against the post eight minutes later.
Lille next face fifth-placed arch-rivals RC Lens away while PSG travel to Stade Rennes.
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