Current world No. 2 Naomi Osaka of Japan would overtake Barty for the top ranking by winning her first match on Wednesday. The 21-year-old opens her Rogers Cup against Germany's Tatjana Maria.
Americans Sloane Stephens, the No. 7 seed, and Venus Williams also were eliminated.
Stephens fell to Marie Bouzkova of the Czech Republic 6-2, 7-5. The 39-year-old Williams, a seven-time Grand Slam champion, lost 6-4, 6-2 to Spain's Carla Suarez Navarro. Suarez Navarro, ranked 26th in the world, snapped a five-match losing streak against the 56th-ranked Williams.
Donna Vekic, Alison Riske, Katerina Siniakova, Zhang Shuai, Iga Swiatek and Ekaterina Alexandrova also advanced to the second round with wins Tuesday.
Also, teenager Bianca Andreescu defeated Eugenie Bouchard 4-6, 6-1, 6-4 in an all-Canadian first-round matchup in her first match since aggravating a shoulder injury at the French Open in May.
Andreescu will face Daria Kasatkina in second round. Kasatkina advanced by upsetting No. 12-seeded Angelique Kerber on Monday night.