Christophe Galtier's side, who beat PSG 1-0 at the Parc des Princes last weekend, had Mike Maignan to thank for as the keeper saved a first-half penalty and made decisive saves.
Metz are 10th on 42 points, seven points behind fellow promoted RC Lens who occupy the last Europa League qualifying spot.
The hosts dominated early on and were awarded a penalty following a VAR review after Jose Fonte appeared to handle the ball in the area.
Maignan, however, stretched to his right to tip away Aaron Leya Iseka's spot kick in the 17th minute.
Lille had their first shot on target in the 40th minute when Celik's shot was easily blocked by Alexandre Oukidja.
Maignan was decisive again five minutes into the second half as he boxed away Leya Iseka's attempt.
Metz, who controlled the midfield, came close again two minutes later, only for Pape Sarr's shot to crash onto the post.
Lille, however, broke the deadlock on the hour when Yilmaz beat the offside trap to collect Renato Sanches's pass before firing past Oukidja for his 10th goal in 22 league games.
Fullback Celik added the second with a powerful right-footed effort one minute from time.
The visitors' defence, the best of the league with 19 goals conceded, then held off Metz as Lille wrapped up their 20th victory of the season.