World No. 15 Monfils beat Australia's Marinko Matosevic 7-6 (7-5) 6-3 6-3 to earn a tie with Brazil's Thomaz Belluci, who eliminated Israel's Dudi Sela.
Simon, seeded 12th, emerged victorious from a five-set marathon against Thailand's world No. 201 Danai Udomchoke to set up a clash with compatriot Julien Benneteau.
Gasquet, the No. 17, will face Kazakhstan's Andrey Golubev after beating Italy's Andreas Seppi 6-3 3-6 6-3 6-1.
Serbian ninth seed Janko Tipsarevic lost the opening set to Russia's Dmitry Tursunov, but scrapped to claim a 5-7 7-6 (14-12) 6-3 6-4 success and a place in round two against Australia's 20-year-old debutant James Duckworth.
American Andy Roddick advanced with a 6-3 6-4 6-1 success over Dutch world No. 53 Robin Haase, and the 15th seed will next play fellow former world No. 1 Lleyton Hewitt of Australia.
The winner of that match will play either Canadian 23rd seed Milos Raonic or Germany's Philipp Petzschner in round three, with Djokovic looming after that.
Japan's 24th seed Kei Nishikoro also went through, and will play either Simon or Benneteau if he beats Australia's 94th-ranked Matthew Ebden.
Spanish 26th seed Marcel Granollers will play Tsonga or Mello if he defeats Portugal's Frederico Gil, while Argentina's world No. 29 Juan Ignacio Chela could face Ferrer if he gets past Spain's Pablo Andujar.
Czech 29th seed Radek Stepanek crashed out 7-5 7-5 6-3 against French world No. 81 Nicolas Mahut, who will likely face Djokovic in round three if he overcomes Japan's Tatsuma Ito.