Britain's loss to Argentina on Friday coupled with other results in the group meant Leon Smith's side went into the final tie at a sold-out AO Arena in Manchester knowing that only a 3-0 victory would send them through.
But it took only an hour and 21 minutes for those hopes to be dashed as Evans was outplayed by former top-10 star Shapovalov, losing 6-0 7-5.
The 15,700-strong crowd - bigger than Wimbledon's Centre Court and a new record for a Davis Cup tie in this country - did their best to inspire the 34-year-old but he could not force a deciding set.
The Lawn Tennis Association has invested significant sums in hosting team events over the past few years but this is now the second time in three years that Britain have gone out in the group stage and shows up the limited options available to captain Smith following Andy Murray's retirement.
Jack Draper should provide him with a strong number one for years to come but, at 34 and slipping down the rankings, Evans' best days are surely behind him, while Cameron Norrie has struggled with injury this year.
Billy Harris, who impressed on his debut on Wednesday but is very inexperienced at the top level, is already 29, and Smith will surely soon be looking to the likes of 23-year-old Scot Jacob Fearnley and last year's Wimbledon junior champion Henry Searle.
Evans was the standout player for Britain 12 months ago in singles and doubles as the team memorably reached the quarter-finals but he has struggled for form and fitness since, and he just could not find a way into the match.
Shapovalov silenced the crowd with an immediate break of serve and it went downhill from there for Evans in a first set where he won only nine points and appeared to give up in the final game.
He dug in at the start of the second set, firing himself up with each point won, and his chance arrived with two break points in the ninth game.
Shapovalov, currently ranked down at 100 after a knee injury but impressive this week, saved them both, though, and ended Evans' resistance with a break in the final game.
The victory means Canada, who were Davis Cup champions two years ago, join Argentina in advancing from Group D to the final eight event in Malaga in November.
Draper showed what the occasion meant to him despite Canada having joined Argentina in progressing from the group by smashing his racket six times after losing the first set to Felix Auger-Aliassime in a 7-6 (8) 7-5 defeat.
The US Open semi-finalist had beaten the Canadian, who is ranked just one place below him at 21 in the world, in both their previous meetings, including in Cincinnati last month in a contest that ended in controversy after Draper was incorrectly deemed to have got to a ball before it bounced twice.
This was an extremely close, high-quality encounter but this time it was Auger-Aliassime who had the edge to secure victory for Canada in the tie and extend Draper's losing record in Davis Cup to one win from four matches.
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