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Federer, who got involved in a lengthy argument with the chair umpire during the ill-tempered third set, converted his first match point and will meet Novak Djokovic for a place in Sunday's final.
Men's title-holder Novak Djokovic eased into the last 16 but elsewhere there were surprises all over Melbourne Park with Williams the biggest victim and Federer fighting back from 4-8 down in a fifth-set tie-break with John Millman.
Borg's Europe have now won the title in each of the three years the team event has been staged.
Team Europe's Nadal was also scheduled to take on Team World's Nick Kyrgios in his singles clash.
Rafael Nadal and doubles partner Stefanos Tsitsipas were beaten by Nick Kyrgios and Jack Sock in the final match of the day as Team World cut down Team Europe's lead to 7-5.
Pella's win over the Canadian took him to his first major quarterfinal, where he will face Spaniard Roberto Bautista Agut.
Reilly Opelka came through 7-5, 3-6, 4-6, 6-4, 8-6 against the 22nd seeded Swiss for the biggest win of his career.
Federer, bidding for his 101st career title and a record sixth Indian Wells crown, received a walkover into the final, where he will play Dominic Thiem.
Australia goalkeeper Maty Ryan earned the Asian Cup holders a quarter-final place with two saves in the shootout to see off Uzbekistan 4-2 on penalties after the match finished goalless following extra time on Monday.
Injury-prone Raonic, who climbed to a career-high ranking of third in 2016 when he reached the Wimbledon final, believes he was rediscovering his best form.