Djokovic, 31, took 84 minutes on Wednesday to advance with seven aces and two breaks of serve while never facing a break point against Toronto-based wild card Polansky, the world number 121.
The ninth-seeded Serb, competing in his first tournament since his Wimbledon triumph last month, won a dominant 83 percent of his first service points and wrapped up the day's opening match minutes before the first of several rain delays.
"I feel very proud for me, myself, and my country".
"I'm pretty comfortable with it - it's good to have shot clock in the tournaments prior to the US Open".
The Australian said his hip injury was still an issue but we would play doubles with Ivan Dodig later on Tuesday. "I've never felt so many emotions after a victory".
Djokovic, a four-time Rogers Cup champion, faded in the decisive set against the flashy 19-year-old, who played with poise and power on a warm, breezy afternoon at Aviva Centre on the York University campus. "I really enjoyed myself out there". "I just tried to stay focused". I had my chances to come back in the third set when I dropped my serve at 15-40. "You have to deal with these kind of situations".
Tsitsipas set up match point with a scorching crosscourt victor before completing the victory in two hours 17 minutes. "I knew I was playing pretty good today".
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'But I remained calm. "I just patiently waited for the moment and it happened". But I think one is good to give away to a top player, but two is, in my opinion, pushing it'.
Nadal will face Stan Wawrinka in the round of 16 after the Swiss rallied to beat Hungarian Marton Fucsovics 1-6 7-6 (2) 7-6 (10).
"He played very well and deserved to win without a doubt", Djokovic said. Dimitrov, who last reached a quarter-final in April, added: "Clearly I'm not playing my best tennis, but I'm finding a way and managing to go through those matches. Just happy to be in the quarterfinals".
Sixth seed Marin Cilic awaits in the quarter-finals following his speedy defeat of Argentina's Diego Schwartzman 6-3 6-2.
"This was the best match of my career", Tsitsipas said.
"Honestly I'm not sure what he said". 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.
Fourth-seeded Kevin Anderson of South Africa and fifth-seeded Grigor Dimitrov of Bulgaria set up another quarterfinal.
Del Potro, who had a bye in the first round, was set to play Robin Haase of the Netherlands on Wednesday.