Szavay recorded an expected 6-4, 6-2 victory over Viktoriya Kutuzova in a first-round match at the clay-court tournament.
Cornet, coming off a tournament win at Hungary last week, was just behind 2-6, 6-2, 3-2 to Rossana De Los Rios when she couldn't continue due to an intestinal problem.
Eighth-seeded Vera Dushevina was also out of the tournament after a 6-3, 6-4 loss to Karolina Sprem.
But third-seeded Carolina Wozniacki swept by Su-Wei Hsieh 6-1, 6-1 and No. 5-seeded Timea Bacsinszky got past Liana Ungur 6-3, 3-6, 6-3.
Also winning first-round matches Wednesday were Tereza Hladikova, Lucie Hradecka, Mariya Koryttseva, Anna-Lena Groenefeld, Yvonne Meusburger and Patricia Mayr.