Goerges needed 2 hours, 47 minutes to make her way past France's Olivia Sanchez 7-6 (7-5), 2-6, 7-5 and secure a second-round berth.
Slovenia's Hercog, meanwhile, outlasted Frenchwoman Pauline Parmentier 6-7 (2-7), 7-6 (12-10), 6-2 on the clay courts of Club Campestre El Rancho.
Other seeded players registering wins Tuesday included No. 5 Carla Suarez Navarro of Spain, who disposed of Slovenian Masa Zec Peskiric 6-3, 6-2, and No. 7 Lourdes Dominguez Lino of Spain, who bested Mandy Minella of Luxembourg 6-1, 6-4.
Sixth-seeded Edina Gallovits-Hall of Romania, however, went down to defeat, 5-7, 6-4, 6-3 at the hands of Russian Nina Bratchikova.