Thiem, the 2016 champion in Acapulco, didn't face a break point in his 75 minutes on court against the 18-year-old Canadian as he set up a last-eight clash with former US Open champion Juan Martin del Potro.
"I'm very happy because everybody knows how good he can play and how dangerous he is," Thiem said of Shapovalov. "I was really on from the first point. Today was almost perfect."
Argentina's del Potro, the sixth seed, needed three sets to get past Spain's David Ferrer 6-4, 4-6, 6-3.
"The first set was very tough, but I played with confidence and after winning the tie-break I was able to do things better," Zverev said.
South Korean Chung Hyeon, a surprise semi-finalist at the Australian Open, didn't face a break point in his 6-3, 6-1 victory over 21-year-old American Ernesto Escobedo.
Chung reached his fourth quarter-final of the season and will next face fifth-seeded South African Kevin Anderson, a 6-3, 6-4 winner over France's Adrian Mannarino.
In women's action, top seed Sloane Stephens saved a match point on the way to a 5-7, 7-6 (8/6), 6-0 victory over Dutch qualifier Arantxa Rus.
"I am happy," Stephens said. "I was match point down, and I was able to come back and fight.
"For not having played for a while, since Australia, it's good progress," added the American, who until this week hadn't won a match since her triumph at Flushing Meadows.
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