Playing in front of a 15,000-strong crowd at Manchester Arena, Shapovalov swept through the first set in 25 minutes, with Evans winning just nine points.
The Briton, 34, finally got on the scoreboard when he held early in the second set, but he failed to make the most of two break points at 4-4.
Shapovalov was able to back up a dominant first set when he struck the decisive blow at 6-5 to take his nation to the knockout stages, which will take place between 19-24 November.
The 25-year-old Canadian, who was ranked as high as 10 in the world after a run to the Wimbledon semi-finals in 2021, has struggled with a knee injury over the past year.
“Sometimes 6-0 is the worst lead in tennis. You can’t improve from that first set,” Shapovalov said after wrapping up victory in one hour and 22 minutes.
“I’m extremely happy with the way I was able to serve. I played really well and just happy I was able to capitalise on my chances today.”
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