Ninth seed Simon dropped the first set on a tie-break but came through to beat Italy’s Fognini 6-7(1) 6-3 6-3, while 12th seed Monfils dropped the second set in his 7-5 6-7(2) 6-3 victory over Germany's Kohlschreiber.
Simon plays Spain’s Guillermo Garcia-Lopez in the second round, with Monfils up against Russia’s Igor Andreev as they compete for a third-round match with Tomas Berdych.
In a late match on Tuesday, 10th seed Juan Martin Del Potro beat Germany's Florian Mayer 6-4 6-2, with Richard Gasquet needing three sets to beat Thomaz Bellucci 4-6 6-4 7-6(5).
Del Potro, continuing his fine comeback after an injury-ridden 2011, faces Mikhail Youzhny next, with Gasquet up against Viktor Troicki.
Other seeds to progress on Tuesday were Spain’s Fernando Verdasco, who cruised to a 7-5 6-2 win over Uzbek Denis Istomin, while Ukrainian Alexandr Dolgopolov beat Pablo Andujar of Spain 7-6(5) 6-4.
Verdasco, the 15th seed, next faces Colombia's Alejandro Falla, while 16th seed Dolgopolov plays Italy’s Andreas Seppi after he beat Michael Llodra of France 7-5 6-4.
In other second-round clashes the in-form Canadian Milos Raonic – who beat world number four Andy Murray in Barcelona – ousted veteran Argentine David Nalbandian 6-4 6-4, while Radek Stepanek of the Czech Republic beat Bernard Tomic of Australia 6-2 7-5.
Raonic faces a prestige clash with Roger Federer, while Stepanek plays David Ferrer of Spain.
Spain’s Marcel Granollers beat Carlos Berlocq of Argentina 7-5 3-6 7-5, but compatriots Albert Montanes and Albert Ramos were beaten 6-3 6-3 by Croatia’s Marin Cilic and 6-3 6-2 by Federico Delbonis of Argentina respectively.
Cilic plays American John Isner in the second round, while Delbonis contests Serb Janko Tipsarevic.
Meanwhile, Stanislas Wawrinka of Switzerland beat Olivier Rochus of Belgium 6-4 7-6(3) to set up a clash with Austria's Juergen Melzer, who beat 13th-seeded Spaniard Feliciano Lopez 3-6 7-6(6) 6-4.
The winner of that clash faces Novak Djokovic in the third round, the world number one having beaten Daniel Gimeno in their second-round clash later on Tuesday.