Hazard, voted the French league's best player last season, scored for a third consecutive league game when he cancelled out Cheick Diabate's first-half opener from the penalty spot.
Lille stayed fourth in the standings with 12 points from seven matches, two points behind leaders Lyon, who travel to Caen on Wednesday.
The northerners, without the suspended Mathieu Debuchy and injured Aurelien Chedjou, were shaky at the back and midfielder Florent Balmont brought down Diabate in the area early on.
Diabate got up to convert the resulting spot kick in the eighth minute but Lille fought back after the break.
Hazard fired past Kevin Olimpa in the 57th minute after a fine one-two with Joe Cole to level the scores.
Bordeaux, who are 11th on seven points, continued to push forward and were awarded another penalty 11 minutes from time after keeper Mickael Landreau fouled Fahid Ben Khalfallah.
This time Diabate smashed the spot kick against the post.
On Wednesday, bottom side Marseille will try to get out of the relegation zone when they bid for a first league win of the season at home to promoted Evian Thonon Gaillard.