Romania's Begu dropped the middle set but recovered to pull out a 6-2, 3-6, 7-5 decision over Croatia's Ajla Tomljanovic.
Spaniard Dominguez Lino, meanwhile, got past two-time former champ and five-time runner-up Flavia Pennetta of Italy 7-5, 6-3.
Pennetta lost to fellow Italian Sara Errani in last year's Acapulco final.
Also posting Day 1 wins Monday were Kazakhstan's Sesil Karatantcheva and Estrella Cabeza Candela of Spain.
Celebrity Families of 2014 [PHOTOS]