Rybakina, the reigning Wimbledon champion, finished with a 6-2 edge in aces and picked on Andreescu's second serve, winning 11 of 14 points.
Rybakina took the first set in 38 minutes, but Andreescu jumped out to a 3-1 lead in the second set. However, Andreescu lost her final three service games as Rybakina closed out the match.
Latvia's Jelena Ostapenko, the 13th seed, and Belarus' Victoria Azarenka, the 15th seed, both won their opening matches in straight sets.
Ukraine's Anhelina Kalinina upset 10th-seeded Veronika Kudermetova of Russia 6-3, 5-7, 7-5, and Romania's Sorana Cirstea edged 11th-seeded Beatriz Haddad Maia of Brazil 4-6, 7-6 (7), 7-5 after saving two match points in the second-set tiebreaker.
Romania's Ana Bogdan, the Czech Republic's Karolina Muchova and Karolina Pliskova, the United States' Lauren Davis and Shelby Rogers and Belarus' Aliaksandra Sasnovich also won their opening matches.
Merida Open Akron
The United States' Caty McNally upset third-seeded Lin Zhu of China 7-6 (3), 6-7 (5), 6-1 in the first round at Merida, Mexico.
McNally overcame serving eight double faults (to three for Zhu) and winning just 27 per cent of her second-serve points, 10 of 37. The match featured 15 service breaks, nine of them on Zhu's serve.
In another US-China matchup, the Chinese player won, as Xiyu Wang eliminated eighth-seeded Alison Riske-Amritraj 7-5, 4-6, 7-6 (1).
Seventh-seeded Elisabetta Cocciaretto of Italy cruised past Switzerland's Viktorija Golubic 6-0, 6-3.
Belgium's Ysaline Bonaventure, Spain's Nuria Parrizas Diaz, Australia's Kimberly Birrell and the United States' Katie Volynets also advanced to the second round.
