Fifth seed Jerzy Janowicz was almost apologetic towards the home crowd after winning 6-2 7-5 against Simon.
“Definitely I played my best match of the season,” said the Pole.
“I knew I had to play my best tennis because Gilles moves incredibly well on the court.”
Janowicz, who lost the semi-final to Gasquet last year, will now met Joao Sousa after the Portuguese defeated third seed Philipp Kohlschreiber 7-6 6-4.
Gasquet dispatched Uzbekistani Denis Istomin 6-3 6-4 in little over an hour to set up a clash with compatriot and top seed Monfils, who came from behind to beat qualifier Steve Darcis 4-6 6-2 6-3.