Last Sunday, Stosur won the fifth title of her career in Osaka, Japan, while Ivanovic lost in the Linz final.
Kuznetsova, a two-time Grand Slam champion, broke twice in the first set. But Vinci offered more resistance in the second, trading breaks twice before the Russian took a 4-2 lead.
The Russian will be playing in her first semifinal this season and her first at a home event since she lost to Serena Williams in 2007.
In the men's tournament, Roger-Vasselin strained his neck during the warm-up and had it treated after the first game.
“From his first service game and down in the match he couldn't serve normally,” Seppi said. “I would never like to win like this but sometimes it happens.”
Seppi will next play Mikhail Kukushkin, who beat Andrey Golubev 6-4, 7-6 (4).
