The fourth-seeded Begu struggled but finally prevailed over Switzerland's Timea Bacsinszky 7-6 (7-3), 3-6, 6-4 in a 3-hour test on the red clay at the Fairmont Acapulco Princess.
Brianti, the No. 8 seed, also had her hands full but recovered from a dropped first set to get by Alize Cornet of France 3-6, 6-3, 6-3.
Also Monday, Melinda Czink of Hungary slipped past Canadian Stephanie Dubois 7-6 (8-6), 6-3 and France's Stephanie Foretz Gacon outlasted compatriot Mathilde Johansson 6-4, 7-6 (7-5).
In another first-round match, Mexican Ximena Hermoso dispatched Austria's Patricia Mayr-Achleitner 7-5, 6-3.
Notable deaths of 2014 [PHOTOS]