Needing only one match win in Manchester to eliminate the hosts and top Group D, Canada got the job done in double quick time as Denis Shapovalov dismissed Dan Evans 6-0 7-5 before celebrating by eating ice cream on court.
"It definitely wasn't easy to play against so many Britons," said Shapovalov, who kept his cool in front of a partisan crowd of around 15,000 fans.
"I've experienced it in the past as well. I just tried to stay focused, tried to stay in my bubble and tried to stay ahead as much as I could because I could feel Dan getting pumped up and the crowd getting pumped up.
"It was definitely starting to turn, so I'm very happy I was able to get it done in straight sets."
Felix Auger-Aliassime gave the 2022 winners an unassailable 2-0 lead by outlasting British number one Jack Draper 7-6(8) 7-5 in a high-quality clash.
Britain then ensured the crowd had something to cheer about as Henry Patten and Neal Skupski cruised past Gabriel Diallo and Alexis Galarneau 7-5 6-4.
The Final 8 knockout stage will be held in Malaga from the 19th November to the 24th November, with a draw taking place this week to determine ties.
Italy, the Netherlands, Spain, Australia, the United States, Germany, Canada and Argentina have qualified.
With world number one Jannik Sinner watching on in Bologna, Italy surged to a 2-0 lead against the Netherlands to top Group A after Matteo Berrettini beat Botic van de Zandschulp 3-6 6-4 6-4 and Flavio Cobolli downed Tallon Griekspoor 7-6(4) 4-6 6-3.
The Dutch did not lose hope and advanced as runners-up when Wesley Koolhof and Van de Zandschulp defeated Simone Bolelli and Andrea Vavassori 7-6(6) 7-5 in the doubles match.
Spain rested world number three Carlos Alcaraz for their tie with Australia but earned a hard-fought 2-1 victory to top Group B in Valencia as Marcel Granollers and Pedro Martinez rallied past Matthew Ebden and Max Purcell 5-7 6-4 6-4.
Pablo Carreno Busta had given Spain the lead with a 2-6 6-2 7-6(3) win over Jordan Thompson but Alexei Popyrin beat Martinez 6-4 6-4 to draw last year's runners-up level.
Earlier, Chile finished their campaign on a positive note in Zhuhai with a 2-1 victory over Slovakia to finish third in Group C, although both teams were out of contention.