Schiavone, the 2010 French Open champion, trailed 4-0 in the first set and 4-1 in the second as she slipped to a ninth first-round exit in 11 tournaments this season.
Svitolina made seven double faults and blew a 5-2 lead in the first set. The Ukrainian teenager then broke for a 5-3 lead in the next set to even the match. But Mayr-Achleitner broke Svitolina three times in the final set.
The top-seeded Hantuchova made the decisive break in the penultimate game to outlast Czech qualifier Renata Voracova 3-6, 6-2, 7-5, the sixth-seeded Peng beat Tunisian wild card Ons Jabeur 6-4, 2-6, 6-1, and Oprandi rallied past Italian lucky loser Anastasia Grymalska 5-7, 6-1, 6-4.
Cabeza Candela beat Chanelle Scheepers of South Africa 6-3, 7-6 (2), and qualifier Arruabarrena downed Urszula Radwanska of Poland 6-1, 6-3. Radwanska still hasn't won a match on the main tour this season.
Voegele fired nine aces in her 6-3, 6-2 defeat of Silvia Soler-Espinosa of Spain, Van Uytvanck ousted Alexandra Cadantu of Romania 6-4, 6-1, and Zanevska outclassed Moroccan teenager Rita Atik 6-2, 6-0.