Yilmaz and fellow Turkish international Zeki Celik scored the goals for Lille against a Nice team who played most of the second half a man short after defender Jordan Lotomba was sent off.
Veteran striker Yilmaz, 35, had been the inspiration with a brace as Lille came from 2-0 down to win 3-2 at Lyon last weekend and he was also the catalyst here as Christophe Galtier's side responded perfectly after PSG had beaten Lens 2-1 earlier to move provisionally to the summit.
Yilmaz scored with a pinpoint low strike into the far corner from 20 metres following a Jonathan David lay-off on 13 minutes, and he also struck the post in the second half.
After Lotomba was shown a second yellow card for a foul on Jonathan Bamba at the start of the second half, Celik volleyed in Lille's second goal in the 56th minute as he latched onto a clearance and volleyed in from just outside the box.
If Lille win their final three games against Lens, Saint-Etienne and Angers, they will be champions for the first time since 2011.
"Three points, a clean sheet, it's good," captain Jose Fonte told broadcaster Canal+.
"It was important to win at home because we hadn't won here for a while, and the battles goes on. We have three difficult games left and we just need to keep fighting."