The French 18th seed looked dominant early on in the Montreal contest, utilising his powerful serve well to wrap up the first set without much hassle.
Norrie stepped up a gear in the second, matching Fils' power to force a tie-break.
Despite creating two set-point opportunities, Norrie was ultimately unable to convert when it mattered and Fils sealed the win in one hour and 43 minutes.
Norrie knocked out third seed Alex de Minaur in the round prior, turning the tables from match-point down by winning 11 out of the next 13 games to beat the Australian 5-7 7-6 (7-5) 6-1 - a victory that saw him become British No 1 again in place of Arthur Fery - but he was unable to build on that win.
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Fils will face Spain's Rafael Jodar in the quarter-finals, after he upset eighth-seed Jiri Lehecka in straight sets.
Brandon Nakashima and Luciano Darderi also progressed to the next round with gruelling three-set victories over Arthur Rinderknech and Nuno Borges, respectively.
World No 1 Sabalenka suffers US Open build-up setback as Alexandrova beats her in Toronto
World No 1 Aryna Sabalenka suffered a setback in her US Open build-up when she was beaten 7-6 (7-3) 4-6 6-4 by Russia's Ekaterina Alexandrova in the fourth round of the Canadian Open in Toronto.
The 16th-seeded Alexandrova fought a hard battle lasting nearly two-and-a-half hours to secure her quarter-final berth.
Sabalenka, a four-time Grand Slam champion and reigning US Open winner, recovered from a set down to force a decider but was unable to halt Alexandrova's aggressive shot-making in the final set.
The Belarusian broke early in the opening set before Alexandrova rallied to force a tie-break, which the Russian won to move ahead.
Sabalenka responded by seizing control of the second set with a break for a 4-3 lead before serving out to level the match.
Alexandrova held her nerve in the decider as Sabalenka's error count rose. Serving to stay in the match at 4-5, the top seed saved two match points, but a double fault on the third handed victory to the Russian.
"I just tried to play every single point as if it was the last one because with her, you don't [get] a lot of chances during the match," Alexandrova said in her on-court interview.
"Honestly, I was trying not to think about the score or anything. Just hit the ball, and that's it.
"I'm super happy that I could win because after the second set, I thought [the chance] was already past me."
Alexandrova will face Ukraine's Elina Svitolina, who defeated American eighth seed Amanda Anisimova 6-2 6-4, for a place in the semi-finals
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