Evans has played 28 Davis Cup ties for Great Britain and was a substitute when they won their last title in 2015.
However he was swept aside in the first set by Shapovalov, who silenced the 15,000-strong crowd at Manchester Arena by taking the opener in just 25 minutes.
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.
Former world number 10 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.
Evans said afterwards he would speak to captain Leon Smith about potentially ending his Davis Cup career.
“I’ve thought about it long and hard. I’ll have to think a little more,” Evans told BBC Sport.
“I love competing for my country, I’ve loved every minute of it but sometimes you’ve got to know when to leave.
“It will be a difficult decision but sometimes it’s nice to give somebody else the opportunity and watch from afar.”