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.
In-form Norrie battles past Carballes Baena in straight sets
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.
In-form Norrie battles past Cilic to set up final showdown with Ramos-Vinolas
Nadal beats Britain's Norrie to ease into Barcelona last four