The 14th seed raced to a two-set advantage, winning the first two sets 6-4, taking a single break in each, as Monfils struggled with a knee injury.
The treatment seemed to work, as Monfils was more competitive in the third set, eventually breaking his compatriot to take the set and get a foothold in the match.
The start of the fourth set was delayed while Simon received treatment for a leg injury, which had hampered his movement around the court in the previous set.
The 28-year-old was visibly in pain throughout the fourth set, which Monfils easily won 6-1.
There was also more work for the trainer in that set, as Monfils needed treatment for a blister on his racquet hand.
Both players were sapped of energy by the fifth set, with neither able to make much of an impact during their service games.
Long rallies also took the toll towards the end of the match as both men struggled to find winners, at one point notching up a 71-stroke rally during the contest.
Simon eventually found himself a break up, and was able to serve out the match to set up a fourth-round tie against Britain's Andy Murray.