Becker survived a barrage of 13 aces from the sixth-seeded Chardy but broke his opponent four times during his two hour and six-minute match.
The 34-year-old Becker will face the winner of Grigor Dimitrov of Bulgaria's match against Joao Sousa of Portugal.
Seventh-seeded Nick Kyrgios required less than an hour and fired in dozen aces to beat Tatsuma Ito of Japan 6-3, 6-2 in his second round match.
Ivo Karlovic, seeded No. 3 in the tournament, faced tough resistance from Nikoloz Basilashvili before clinching a 7-5, 4-6, 6-3 victory and will face Kyrgios in the quarterfinals.
Mischa Zverev of Germany will face the second-seeded Feliciano Lopez in the second round after his hard fought 7-6 (2), 3-6, 6-3 win against Rajeev Ram of the United States.