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.
"Today I played really well in the first set with some slight mistakes in the second, which I managed to overcome," Kuznetsova said. "I hope I'll be able to boost my performance in the next matches as well."
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 second semifinal Saturday, Anastasia Pavlyuchenkova will face fifth-seeded Simona Halep.
Pavlyuchenkova eliminated Klara Zakopalova 6-0, 6-4, while Halep routed Alisa Kleybanova 6-1, 6-1.
In the men's tournament, Gasquet broke early in the first set and took a 4-0 lead in the second to eliminate the last Russian left in the draw.
"I made an early break and it was very important as I was in control after that," Gasquet said. "In general I played well and a little bit more aggressive than yesterday, I served well and didn't miss so much."
The Frenchman is looking for the eighth spot at the ATP finals in London next month. But he suffered an early exit at Shanghai last week and dropped to 10th in the rankings, hampering his chances of qualifying.
Gasquet will next play Croatian veteran Ivo Karlovic, who advanced to his second semifinal of the season by winning eight consecutive games and landing 11 aces to rout Russian teenager Karen Khachanov 6-4, 6-0.
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).