Hazard, 27, became only the 10th player to reach that landmark when he collected Mateo Kovacic's pass, took the ball around goalkeeper Ben Foster and calmly finished in first-half injury time.
Chelsea's lead only lasted two minutes as Roberto Pereyra volleyed past Kepa Arrizabalaga from the edge of the 18-yard box to level.
But Hazard grabbed his second of the match - his 101st Chelsea goal in 322 appearances - in the second half when he converted a penalty after Foster had bundled him over.
Belgium forward has been repeatedly linked with a move to Real Madrid and in October said it was his "dream" to join the Spanish side, although two days after that he ruled out a move to the Bernabeu in the January transfer window.
"To score 101 goals with this amazing club is something I will never forget, but now the fans, staff and players want more," said Hazard, who helped Chelsea win the Premier League title in 2015 and 2017.
"I want to score more for this club and then try to be a legend like Frank Lampard, Didier Drogba and John Terry."
His goals took the Blues back up to fourth after they had dropped below Arsenal following the Gunners' 1-1 draw at Brighton earlier on Wednesday.
"Eden Hazard can play in every position," added Blues boss Maurizio Sarri. "He played very well with two goals and could have had two or three assists. He can play with team-mates very well."