Suarez-Navarro will play AgnesSzavay in the third round, with the Hungarian 27th seed progressing after a 6-3 6-2 victory against Taiwan's Chan Yung-Jan.
Hungary's AgnesSzavay progressed to the second round after Italian fourth seed Flavia Pennetta was unable to complete their match despite winning the first set.
Seeds Agnieszka Radwanska, Flavia Pennetta, Alisa Kleybanova, AgnesSzavay, Sybille Bammer and Aleksandra Wozniak also advanced, but Alize Cornet, Dominika Cibulkova, Zheng Jie, Ai Sugiyama and Sara Errani failed to progress.