Former world No. 5 Tsonga, who is currently ranked 210 after a spell out due to knee surgery, overcame No. 4 seed Simon 6-2, 7-6 (3) in one hour, 36 minutes.
Berdych, who lifted the title in the south of France in 2012, defeated the No. 5 ranked Frenchman in just 59 minutes and is looking to win his first title since the Shenzhen Open in 2016.
In the remaining round of 32 ties, Marcos Baghdatis beat Ruben Bemelmans 6-2, 6-4 to set up a clash with No. 1 seed Lucas Pouille, No. 5 seed Philipp Kohlschreiber was beaten 6-4, 6-3 by Radu Albot while Ilya Ivashka beat Thomas Fabbiano 4-6, 6-4, 6-3.