Medvedev, ranked No. 2, lost to Rafael Nadal in the final of the Toronto event in 2019, the last year it was played. Nadal withdrew from the tournament on Tuesday with a foot injury.
The 21-year-old Andreescu, who won the US Open in 2019, was playing her first match since a first-round exit at Wimbledon in late June.
“My goal isn’t to be on the court that long,” Andreescu said. “I want to be able to stick to my game plan the whole entire match and I feel I haven’t been doing that in my matches a lot.
“That comes with experience and playing matches, and I haven’t been playing many matches.”
Also in Montreal, Coco Gauff defeated Anastasija Sevastova, while Johanna Konta — returning from a tough bout with COVID-19 — defeated Zhang Shuai, who retired in the second set. Sara Sorribes Tormo beat Carol Zhao.
On the men’s side, third-seeded Stefanos Tsitsipas beat France’s Ugo Humbert 6-3, 6-7 (13), 6-1, while Casper Ruud of Norway topped veteran Marin Cilic 6-3, 3-6, 6-3. American John Isner won his opening-round match 6-4, 6-1 over Alejandro Davidovich Fokina.