The World No. 26, who enjoyed a run to the last four at the BNP Paribas Open last month, played consistently and aggressively from the baseline, saving two set points in the second set to down Norrie 6-3, 7-6(3) in one hour and 33 minutes.
Fritz has now won 11 of his past 13 matches, having reached the final in St. Petersburg last week. The American earned his third Top 10 win of the year in the second round in Paris when he upset World No. 6 Andrey Rublev.
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With his victory, Fritz has levelled his ATP Head2Head series against Norrie to 4-4. The 24-year-old had never been beyond the second round in two previous main-draw appearances in Paris, but will next face Novak Djokovic for a place in the semi-finals after the World No. 1 received a walkover from Gael Monfils.
Norrie has earned a career-best 50 tour-level victories this season and is 11th in the FedEx ATP Race To Turin (2,945 points), battling to clinch one of the final two remaining singles spots available at the season finale, to be held at the Pala Alpitour from 14-21 November.
However, the Briton now faces a tough battle to qualify. Seventh-placed Casper Ruud (3,185) will secure his spot in Turin with victory over Marcos Giron later on Thursday, while eighth-placed Hubert Hurkacz (3,135) can stamp his ticket with a win against James Duckworth on Friday.
Hurkacz Improves Turin Hopes, Plays Duckworth In Paris QFs
In a strong first set, Fritz struck the ball with power and depth on his forehand to dictate, forcing Norrie deep behind the baseline. The 24-year-old found the decisive break in the sixth game to lead, before sealing the set on serve.
The American continued to push in the second set but Norrie showed great resilience, fending off both break points he faced. However, the 26-year-old was unable to convert two second set points on Fritz’s serve at 5-4, 40/15 and 40/30 as they moved to a tie-break. Fritz pulled away in the tie-break as he found his best tennis, securing his victory with one of the 25 winners he hit in the match.