Anderson, losing finalist last year at Flushing Meadows, fired 11 aces and clubbed 31 winners in his second five setter of the week after going the distance in his first-round battle with American Ryan Anderson in Monday's heat.
In cooler conditions on Friday, Anderson's serve was, as usual, his most potent weapon but it was his defence that proved the difference.
The towering South African saved seven out of nine break points he faced and forced Shapovalov into 77 errors, much to the dismay of the vocally pro-Canadian crowd.
Talented teenager Shapovalov, the 28th seed, was the sloppier of the two and topped the unforced errors count while also stumbling into eight double faults.
The roof closed on the newly built Louis Armstrong Stadium for the first time during the second set, but it did not appear to disrupt the momentum of either player.
Anderson heaped praise on his opponent, who is 13 years his junior, calling Shapovalov an "unbelievable competitor" during a post-match broadcast interview.
"That was an amazing match," the 32-year-old said. "You're definitely going to be seeing him in the years to come."
Shapovalov was similarly deferential, telling reporters after the match that Anderson "played unbelievably big".
"I'm really happy I was able to compete out there with him," the 19-year-old said. "It was a close match. I had a lot of chances to break back, wasn't able to do it today."
Anderson, who has reached the final at two of the last four Grand Slams, leads his fourth-round opponent Thiem 6-1 in their head-to-head but fell to the 24-year-old in May in the Madrid Masters semi-finals.
"Going to have my work cut out for me," said Anderson. "Really going to have to focus on my game, continue doing what I'm doing."
(Reporting By Amy Tennery,; Editing by Nick Mulvenney)
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