Schiavone broke for a 4-3 lead in the first set and led 4-1 in the second before Cornet rallied to tie at 4-4. But the Frenchwoman, who saved 16 of 21 break points in the match, then dropped serve in the penultimate game.
Scheepers won six straight games to take the first set and lead 1-0 in the second against Mladenovic, who won the French Open junior title in 2009.
Lourdes Dominguez Lino of Spain also advanced by beating defending champion Kiki Bertens of the Netherlands 6-4, 4-6, 6-3. The unseeded Spaniard will meet Mandy Minella of Luxembourg, who edged Silvia Soler-Espinosa of Spain 6-3, 3-6, 6-3.