Kevin Monnet-Paquet equalised in added time to salvage a point for the visitors.
Cole, who has been in impressive form in his four Ligue 1 appearances, scored with a superb 20-metre strike but once again Lille failed to keep a clean sheet.
They have drawn their last four games in all competitions, blowing the lead three times in the last five minutes, and now sit in sixth place with 13 points from eight games.
Javier Pastore once again proved his worth as the Argentine midfielder netted twice to help earn Paris St Germain a 3-0 win at Montpellier and joint top spot in Ligue 1.
Pastore, who moved to PSG from Palermo for a French record 42 million euro fee, added to Kevin Gameiro's opener with a stunning volley from Siaka Tiene's cross and made it 3-0 10 minutes from time after a defensive blunder.
Gameiro opened the scoring agains the Ligue 1 pacesetters with a fine volley five minutes before the break but it was Pastore's class that caught the eye.
In six league appearances, the Argentina international has already scored four goals and set up two, adding much-needed technical ability to his team's midfield.
PSG have 17 points from eight games, level on goal difference with Lyon who beat visitors Bordeaux 3-1 thanks to a Bafetimbi Gomis double.
France striker Gomis put Lyon 2-0 up by halftime, with Michel Bastos adding a third after the break as Lyon moved into first place with 17 points from eight matches.
Gomis latched onto a perfect Michel Bastos cross to put the home side ahead after eight minutes and was on the spot again in the 33rd when he capitalised on fine work by Clement Grenier on the right flank to fire past Cedric Carrasso.
Man-of-the-match Bastos whizzed into the box and flicked the ball past Carrasso to put the result beyond doubt just past the hour mark, with Anthony Modeste scoring a consolation penalty for the visitors spot in the 86th.
Toulouse, who beat Nancy 1-0 thanks to a second-half goal by Turkish striker Umut Bulut, are third on goal difference, one point above Montpellier.
Marseille, down to 10 men after Rod Fanni was sent off on the stroke of halftime, also had to settle for a 1-1 draw against Valenciennes after Jose Saez struck three minutes into added time to cancel out Souleymane Diawara's first-half opener.
Marseille, having won only one league game this season, are languishing in 13th spot on seven points.