Goerges bested Belarusian Olga Govortsova 7-5, 6-4 on the green clay at Family Circle Tennis Center in Charleston, S.C.
Schnyder, meanwhile, didn't fare as well. The lefthanded Charleston finalist in 2002 and 2006 fell to South African Chanelle Scheepers in straight sets, 6-4, 6-4.
Also Monday, American Vania King downed 16th-seeded Russian Vera Dushevina 6-4, 7-6 (9-7), and Jill Craybas of the United States prevailed over fellow American Shelby Rogers 3-6, 6-2, 6-0.
Elsewhere in Charleston, Briton Heather Watson was ousted by American Christina McHale 5-7, 6-3, 7-6 (7-3).
Other Day 1 winners included Swede Sofia Arvidsson, Czech qualifier Eva Birnerova, Russians Elena Vesnina and Evgeniya Rodina, and Japan's Ayumi Morita.
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