Chasing, more at ease on his forehand than on his backhand, Fils only found the opening at the end of a poorly started tie-break (0-2). He had a bit more trouble making the difference in the second set against an opponent seeking a rebound, who dropped from 20th in the ATP rankings to 60th in a year.
Fils showed his intention to do well from the first strokes of the racket.
Earlier, three other French players had already booked their tickets for the round of 16.
This victory is also reassuring on the physical front, a week after his mid-match retirement in Basel.
It is the second piece of good news of the day for Arthur Fils.
He opened the match with a service winner, earned the first break as early as Machac's first service game and led 3-0 in just seven minutes. After sailing through all his service games while hanging on to the Czech on his own, the Frenchman logically took the lead after forty minutes (6-3).
The 22-year-old Frenchman left his opponent no opening at the Arena de Nanterre.