The French 18th seed looked dominant early, utilising his powerful serve well to wrap up the first set without much hassle on Saturday (Sunday AEST).
Norrie stepped up a gear in the second, matching Fils' power to force a tiebreak.
Despite creating two set-point opportunities, Norrie was ultimately unable to convert when it mattered and Fils sealed the win in one hour 43 minutes.
Fils will face Spanish teenager Rafael Jodar in the quarter-finals, after he earlier in the day upset eighth-seed Jiri Lehecka in straight sets.
The 19-year-old 20th seed, who has one ATP Tour title to his name, won 6-3 6-3 in 76 minutes.
Brandon Nakashima and Luciano Darderi also progressed to the next round with gruelling three-set victories over Arthur Rinderknech and Nuno Borges, respectively.
"I think the level is continuing to grow with each match," world No.31 Nakashima, who reached his first Masters 1000 quarter-final at the sixth time of asking, said.
"Every day it is a new opportunity out here. So (I'm trying) to make the most of it. Super, super happy that it's all paying off."
With AAP.
8°C