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British number one Johanna Konta was leading two-time Grand Slam champion Victoria Azarenka 6-3 3-0 when her Rogers Cup second-round match was halted by rain in Montreal.
Play will resume on Thursday and Konta, 27, could face world number five Elina Svitolina later in the day if she wins.
Angelique Kerber's first match since she won Wimbledon ended in a 6-4 6-1 defeat by France's Alize Cornet.
Meanwhile, Maria Sharapova thrashed fellow Russian Daria Kasatkina 6-0 6-2.
Play was also suspended as world number one and French Open champion Simona Halep faced Russia's Anastasia Pavlyuchenkova, with the latter leading 4-3.
Konta, the world number 43, broke the Belarusian in the seventh game and again in the ninth, to take the opening set in 40 minutes.
Former world number one Azarenka, 29, who is returning after having a son, was broken early in the second, before the game was stopped.
World number four Kerber, 30, who beat Serena Williams to claim her third Grand Slam title in July, made 32 unforced errors in her second-round match.
Cornet, ranked 30 places below German Kerber, won in 85 minutes.
"It was my first match after three weeks," said Kerber. "It's not always so easy to change the surface. I think I need a little bit more time to get used to hard courts again."
Sharapova hit 25 winners in her win, with the five-time Slam champion, 31, needing 76 minutes to see off 21-year-old Kasatkina, who is considered one of the rising stars of the women's game and ranked 12th in the world.
US Open champion and third seed Sloane Stephens also made the third round with a 6-0 6-2 win against Canadian Francoise Abanda 6-0, 6-2.
In the men's competition, which is played in Toronto, world number one Rafael Nadal is through after beating Frenchman Benoit Paire 6-2 6-3 in 74 minutes.
The Spaniard will play Stan Wawrinka in the third round after the Swiss rallied to beat Hungarian Marton Fucsovics 1-6 7-6 (7-2) 7-6 (12-10).
Wimbledon champion Novak Djokovic kept his bid for a fifth Rogers Cup title alive with a 6-3 6-4 win over Canadian wildcard Peter Polansky.
The world number 10 is playing his first tournament since winning a 13th Grand Slam title last month and in the third round faces Greek teenager Stefanos Tsitsipas, who upset Austrian seventh seed Dominic Thiem 6-3 7-6 (8-6).
Defending champion Alexander Zverev is through with a 6-4 6-4 win over Bradley Klahn, while Wimbledon runner-up Kevin Anderson, Grigor Dimitrov, Marin Cilic, John Isner, Robin Haase and Karen Khachanov also progressed.
