Cilic beat 11th-seeded Diego Schwartzman of Argentina 6-3, 6-2.
He rounded off his afternoon with back-to-back winners to send Djokovic, the victor of 30 Masters titles, out on a first match point.
South Africa's Anderson, who Djokovic beat in this year's Wimbledon final, saw off Belarus' Ilya Ivashka 7-5 6-3.
World No 1 Nadal battled past Wawrinka 7-5 7-6 (7-4) to reach the quarter-finals and stay on course for his fourth Canadian trophy at the Masters 1000 event.
"I'm happy to see Stan playing well again". But even the Andy one, I was a little surprised by that one just because he was entered on his protected ranking, and the Federation made a decision to help him out so he could pull out with his protected and get a wildcard entry so he could save it for another tournament. It was all square on 4-4 when Tsitsipas attempted to break the world number ten's service, however Djokovic endured and led the set to a tie-break, which he eventually won, to bring the game to a draw (6-7).
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Anderson posted a 6-2, 6-2 victory over fifth-seeded Grigor Dimitrov of Bulgaria in the early match.
At 30-40, Zverev floated his second serve well long to end the match in two hours 27 minutes. "If we would have played best-of-five, that's what the score should have been".
"I feel very proud for me, myself, and my country". "I just played not that great". "People seemed to love it, loved the show and everything". He rose to prominence a year ago with a stellar run at the Rogers Cup in Montreal that ended with a semifinal loss to eventual champion Alex Zverev. Italy's Fabio Fognini, the No. 14 seed coming off a victory at Los Cabos, fell out with a 6-3, 7-5 (14) loss to Canada's Denis Shapovalov.
Second-seeded Henri Kontinen of Finland and Australian John Peers dispatched Americans Sam Querrey and Rajeev Ram 7-6 (7), 6-3 in the late doubles match.
Garcia proved too strong for Sharapova with a 6-3, 6-2 win and will take on top seed Simona Halep of Romania in the quarter-finals of the Women's Tennis Association Premier 5 tournament. Zverev beat Roger Federer in last year's singles final at Montreal.