Norrie, ranked 102 in the world, only played his first professional match on red clay earlier in the year in the Davis Cup when he beat Roberto Bautista Agut.
The 22-year-old, who has yet to drop a set in Lyon, will now face Mikhail Kukushkin of Kazakhstan or France's Gilles Simon in the semi-final.
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Top seed Dominic Thiem remains in the other side of the draw.
"I'm so stoked with my performance today," said Norrie after his first win against a top 10 opponent.
"In the past, I haven't been so good at playing guys with big serves.
"I remember watching Isner when I was younger. He was a big idol of mine. It's crazy to be competing against guys like that now."
Norrie is now expected to rise into the top 100 and possibly as high as 85.
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