Yusuf Yazici and Weah scored either side of the interval to put Lille on 32 points from 15 games before second-placed Lyon take on Brest and third-placed Paris Saint-Germain, who are one point further adrift, entertain Lorient later on Wednesday.
The result left Dijon, who have won only one game this season, on nine points, four from safety.
Lille went ahead in the 19th minute when Yazici found the back of the net with a low shot from Zeki Celik‘s cut-back pass.
They came close to doubling the tally before the break, only for Jonathan Ikone to miss the target with an attempted lob.
Dijon rarely threatened but Lille’s Jose Fonte almost beat his keeper Mike Maignan with a poor back pass on the hour, keeping the visitors on their toes until Weah wrapped it up with a half-volley three minutes into stoppage time.