Tsurenko took down the seventh-seeded Spaniard 2-6 6-4 6-4 in the early match, which was a preview of the challenges faced by top seeds during the day.
World number one Simona Halep fell down a set but rallied against Australia's Ajla Tomljanovic, and was leading 4-6 6-3 4-3 when her match was suspended due to rain.
Second seed and Australian Open champion Caroline Wozniacki lasted just one set against Kiki Bertens before retiring due to a left knee injury.
The match was delayed by rain for over three hours and Wozniacki retired shortly after Bertens captured the opening set in 62 minutes.
"It's never nice to win when someone has to retire," Bertens said. "I felt she was struggling a bit with her serve. She seemed to struggle with her movement as well."
Fourth seed Angelique Kerber also had a difficult outing, having to battle back to outlast Anastasia Pavlyuchenkova 4-6 7-5 6-4.
Muguruza was competing in her first tune-up event for the Aug. 27-Sept. 9 U.S. Open, having been sidelined due to an arm injury that forced her to withdraw from a pair of tournaments.
It was her first match since losing in the second round at Wimbledon.
"I'm happy that I didn't feel pain," she said. "I competed. It didn't go my way. I'm going to take that positive and keep training for US Open."
Next up for Tsurenko is a clash with Ekaterina Makarova after the Russian topped Frenchwoman Alize Cornet 6-2 6-0.
Joining Tsurenko in the third round was U.S. Open champion Sloane Stephens, who enjoyed a routine 6-3 6-2 win over German Tatjana Maria.
Stephens, who finished runner-up to Halep in Montreal on Sunday, continued her fine form on the North American hard courts and controlled the tempo throughout her 71-minute match.
Stephens will next face Elise Mertens after the 15th-seeded Belgian earned a 3-6 6-2 7-6(1) win over Swedish qualifier Rebecca Peterson.
Also advancing to the third round was Belarusian Aryna Sabalenka, who upset Czech ninth seed Karolina Pliskova 2-6 6-3 7-5, and Australian Ashleigh Barty, a 7-5 6-3 winner over Estonia's Kaia Kanepi.
American Amanda Anisimova had a 5-4 lead on Croatia's Petra Martic in a match that was also halted because of rain.
(Reporting by Frank Pingue in Toronto, editing by Pritha Sarkar and Sudipto Ganguly)