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Thiem was handed a bye to this stage and fell at the first hurdle on a day of upsets, the world number four sending down four double faults and losing serve three times as he lost 6-3 6-2.
Norrie now has 10 victories on clay in 2021, compared to seven before this year, and will face qualifier Arthur Rinderknech – a surprise 6-7(7) 6-2 7-5 winner against Jannik Sinner – in the quarters.
"I'm so pleased to win. It's the biggest win of my career and my highest-ranked win," Norrie, who had never previously beaten a top-five opponent, said in his on-court interview.
"It's such a beautiful day in Lyon. I couldn't be happier to get the win today and to get another match on the clay before Roland Garros."
Third seed Diego Schwartzman also exited the competition on Thursday with a 6-3 7-5 defeat to Richard Gasquet, who eased over the line despite letting two match points slip.
Stefanos Tsitsipas is now the strong favourite to triumph in France after starting his campaign with a convincing 6-1 6-4 victory over Tommy Paul.
Yoshihito Nishioka upset Gael Monfils 4-6 6-3 7-6(2) in an impressive comeback victory and now awaits Tsitsipas in the last eight.
At the Geneva Open, rising star Casper Ruud kept his momentum going with a 2-6 6-1 6-4 win over Dominik Koepfer to make it 12-4 on clay this season.
Ruud recovered from a sloppy first set by holding his serve in the following two sets to reach his fourth successive clay court semi-final on the ATP Tour.
Next up for the third seed is a meeting with Pablo Andujar, who beat youngster Dominic Stephan Stricker 4-6 6-4 6-4.
Elsewhere, Laslo Djere overcame Fabio Fognini 6-3 6-7(2) 6-1 to reach the quarter-finals and will face Denis Shapovalov later on Thursday after the Canadian recovered from a set down to beat Marco Cecchinato 6-7 7-5 6-1.
The winner of that match will meet Pablo Cuevas, the Uruguayan having ended Grigor Dimitrov's run with a battling 7-6 6-3 win.