Hingis rolled to a 6-2, 6-3 win over Tiantian Sun while Szavay was a 7-5, 6-3 winner over Marina Erakovic.
In other matches Tuesday, eight-seeded Eleni Daniilidou swept past Olga Govortsova but No. 7 Anabel Medina Garrigues fell, 6-4, 6-4, to Akiko Morigami. Also, Shuai Peng knocked out Yi-Fan Xu, 6-1, 6-4, and Abigail Spears topped Zi Yan, 7-5, 6-3.
Spear's second-round opponent is top-seeded Svetlana Kuznetsova, who helped Russia to the Federation Cup title last weekend. Kuznetsova is the defending China Open champion.
2014: The Year in Music [PHOTOS]