Stakhovsky upset seventh-seeded Joao Sousa of Portugal 6-4, 6-3, while Cornet ousted Ukrainian teenager Elina Svitolina 7-5, 6-3.
Stakhovsky won seven consecutive games to rally from 4-1 down in the first set and take a 2-0 lead in the second. He broke Sousa again in the ninth game, sealing the win with a drop shot on his second match point.
"I started a bit nervous and experienced certain problems in the beginning," Stakhovsky said. "But later in the first set I managed to bring my concentration back and dominated the play for the rest of the match."
Cornet broke in the 12th game of the first set and took a 4-0 lead in the second.
"I didn't expect to play this level today because I just came back from injury," Cornet said. "But surprisingly I played really good today. I think I was really solid especially in the most important moments and that's why I won."
Earlier, top-seeded Angelique Kerber, coming off her third career title in Linz, Austria, on Sunday, withdrew with an abdominal strain.
Richard Gasquet, who is looking for a spot in the season-ending ATP World Tour finals in London for the second straight season, leads the men's field.