Yusuf Yazici and Weah scored either side of the interval to put Lille on 32 points from 15 games as second-placed PSG, who are one point adrift, beat Lorient later on Wednesday. Third-placed Lyon conceded a last-gasp goal in a 2-2 home draw against 10-man Brest.
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.
The score for Weah, son of Balon D'Or winner George Weah, is his second this season after a goal against his former side Celtic in Europa League group stage action. He missed much of last season with hamstring injuries following a move from PSG and a loan stint with the Scottish club.
"Timothy is coming off a difficult season and has not had a lot of playing time. He works a lot and is very respectful of decisions," Lille manager Christophe Galtier said. "He is determined at training ... we all know his potential, we see it every session."