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.
Torro-Flor cruised past fourth-seeded Bojana Jovanovski of Serbia 6-2, 6-2, and the fifth-seeded Muguruza did not face any break points in her 6-2, 6-2 win over Julia Glushko of Israel.
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.
Stefanie Voegele of Switzerland, Alison Van Uytvanck of Belgium and Ukrainian qualifier Maryna Zanevska also advanced to the second round.
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.