Ivanovic, who won the title in Monterrey last year, was a 6-1, 7-6 (7-4) winner over Nicole Vaidisova, while the second-seeded Errani blitzed past Lauren Davis, 6-0, 6-3, on the hardcourts of the Sierra Madre Tennis Club.
Third-seeded Caroline Garcia, the runner-up at last week's Mexican Open, downed Karin Knapp, 7-5, 7-6 (7-5), and fourth-seeded Timea Bacsinszky needed three sets to send Yanina Wickmayer home in a 7-6 (10-8), 3-6, 6-4 victory.
Rain forced the postponement of four first-round matches from Monday. Fifth- seeded and three-time champion Anastasia Pavlyuchenkova was scheduled for one of those matches, and she overcame a slow start in her 4-6, 6-3, 6-0 win over Anna Schmiedlova.
Seventh-seeded Daniela Hantuchova and eighth-seeded Magdalena Rybarikova both survived their first-round tests, while Tuesday's biggest upset saw Vera Zvonareva pick up one of her most impressive wins in years as she knocked off sixth-seeded Alison Riske, 6-4, 6-0.
Among the others moving on were former French Open champ Francesca Schiavone and Timea Babos, who won this event in 2012. Urszula Radwanska, Lesia Tsurenko, Pauline Parmentier, Kristina Mladenovic, Johanna Larsson and Polona Hercog also advanced to the second round.
This year's champ will earn $111,389.