The 20-year-old Briton remains on course to break into the top 50 ahead of his US Open debut on the back of an impressive run which also saw him make the last eight in Montreal, beating the likes of Stefanos Tsitsipas and Gael Monfils.
Draper, who had reached the semi-finals on home turf in Eastbourne earlier this summer, swiftly took control after moving 5-0 ahead in the opening set against Thiem, who is continuing to try to rediscover his own form after close to a year out with a wrist injury.
Thiem improved in the second set, but also hit six double faults as he tried to put Draper on the back foot.
However, the Briton, currently ranked 55 in the world, eventually got a crucial break in the ninth game to go on to close out the match 6-1 6-4.
Draper, who will face Switzerland's Marc-Andrea Huesler for a place in the semi-finals, said: "I started confidently, I started well.
"I was hitting through the ball, felt good in the conditions. It was a difficult start because I felt like everyone was supporting Dominic.
"I knew that I needed to come out confidently and sharp - and that's what I did."
British number four Draper, who did not face a break point all match, added on the ATP Tour website: "I was really good behind serve, really solid.
"In the second set it was tough because he was coming up with some amazing shots.
"It was a real honour to be on court with Dominic. He's an amazing player."