Federer, who currently occupies the sixth qualifying spot, can strengthen his position with each victory at his hometown event, which he has won five times.
The third-seeded Federer plays his second-round match later Wednesday against Denis Istomin of Uzbekistan.
Juan Martin Del Potro of Argentina, the top-seeded defending champion, followed Gasquet on court in a first-round match against Swiss wild card Henri Laaksonen.
Del Potro is already the only seed remaining in his half of the draw, which includes the 55th-ranked Llodra.
Paul-Henri Mathieu of France advanced to face Llodra next by beating Denis Kudla of the United States, 6-4, 7-5.