Qualifier Kirsten Flipkens, No. 251 in the world, ousted the top-seeded and world No. 5 Stosur 7-6 (9-7), 6-3. Flipkens pounded out nine aces and won nearly 70 percent of the points on serve.
Flipkens was particularly effective in the second set, losing five points over five service games.
Kateryna Bondarenko defeated Errani, the No. 2 seed, 5-7, 6-3, 6-4 by piling up eight breaks. She had four breaks in the second set in leveling the match and converted both her break chances in the third to advance.
Errani defeated Stosur in the semifinals of the French Open less than two weeks ago.
No. 3-seeded Dominika Cibulkova stopped Alexandra Cadantu 6-2, 6-2 in 65 minutes but fifth-seeded Maria Kirilenko was eliminated, losing to Francesca Schiavone 6-4, 3-6, 7-5.
Also Monday, Monica Niculescu beat Barbora Zahlavova Strycova 7-6 (8-6), 6-3; Shahar Peer stopped Akgul Amanmuradova 6-2, 6-4; Arantxa Rus advanced over Pauline Parmentier 6-3, 6-4; Urszula Radwanska rallied for a 4-6, 6-1, 6-4 win over Ksenia Pervak and Irina-Camelia Begu raced past Iveta Benesova 6-4, 6-1.