Marseille, however, continued their miserable start to the season with a 1-0 home defeat against Rennes.
The 2010 champions, who have drawn three and lost two games, lie 17th, eight points off the pace.
Jires Kembo-Ekoko netted the winner for Rennes in the 76th minute after benefiting from some shaky defending.
Lyon, who have 11 points from five games, and lead second-placed Lille by one point.
St Etienne were down to 10 men in the 35th minute after defender Loris Nery was shown a straight red card for a rough tackle on Franck Beria.
Lille were rewarded for their domination 10 minutes into the second half, when Belgian midfielder Hazard slalomed through the St Etienne defence before beating Stephane Ruffier.
Hazard rushed down the left flank in the 73rd minute and unleashed an angled shot that went under the bar to put the visitors ahead.
Cole then whizzed down the right flank, through a couple of defenders, before setting up Ludovic Obraniak for the third goal.
"I'm happy to be here from the moment I stepped on the training ground," Cole told French TV channel Foot Plus.
"Hazard is a very, very special player and I look forward to forming a great partnership."