The second-seeded Simon had nine aces and saved four break points in his 6-3, 6-2 win against Flavio Cipolla of Italy, while the fourth-seeded Gasquet broke Nikolay Davydenko three times in a 6-3, 6-4 victory.
Even though Davydenko is no longer ranked in the world's top 50, Gasquet was wary of the former top-five player.
"He's not the No. 3 he was a few years ago but he can still dazzle sometimes, and you can easily find yourself a break down against him," Gasquet said. "So it was important to hold my serve."
Davydenko led 40-15 on serve in the 10th game of the second set, but Gasquet took a forehand early to pass Davydenko, who next hit an unforced error and watched as Gasquet dispatched a volley to get to his first match point.
Davydenko saved that with his fifth ace. Gasquet wasted his next chance when a hurried backhand pass hit the net, then double-faulted on his third match point and watched as Davydenko saved another with a clipped forehand pass.
Gasquet, who reached the fourth round at the Australian Open, sealed victory on his fifth match point after a long baseline rally ended with the Russian slicing a forehand well wide.
"I'm glad he missed that as it was becoming harder to get my first serve in," said Gasquet, who has slipped from a ranking of seventh in 2007 to 16th. "I've recovered from Australia now and am starting to sleep properly, so everything's fine."
Rufin hit 16 aces compared to 13 for Lopez, who lost his serve twice in the second set after the big servers swapped a break each in the first.