Britain's Cameron Norrie lost in the Canadian Open first round, while compatriots Johanna Konta and Harriet Dart both advanced.
Konta, 30, lost the first set 6-4 to China's Shuai Zhang but was winning the second 5-2 when her opponent retired.
World number 172 Dart, 25, beat Canada's Leylah Fernandez 7-5 7-6 (7-4) and plays Bianca Andreescu next.
Konta will face Ukranian third seed Elina Svitolina in the second round.
The British number one is playing for the first time since having to withdraw from Wimbledon when a member of her coaching team tested positive for Covid-19.