Mahut broke last year's runner-up twice in each set. The Frenchman won two titles in the summer and is enjoying his best season.
The eighth-seeded Mayer did not face a break point in his 6-3, 6-2 victory over Lukas Lacko of Slovakia, while Becker outlasted French wild card Albano Olivetti 6-7 (8), 6-3, 6-1.
Olivetti rallied from 6-2 down in the tiebreaker to win the first set.
But Becker broke Olivetti twice in the next set to even the match. The German took leads of 3-0 and 5-1 in the third set.