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.
Boston schools pull out free condoms over wrapping complaints
Ray Liotta sues skin care company over use of likeness