Ivanovic beat Vesna Dolonc 6-4, 6-1, while the top-seeded Stosur defeated Klara Zakopalova of the Czech Republic 6-1, 6-3.
Stosur won six consecutive games in the first set but trailed 3-1 in the second set. She recovered with a break in the sixth game and later served out the match at love.
The 2011 U.S. Open champion from Australia will play the 2008 French Open champion in the semifinal on Saturday.
Earlier, Caroline Wozniacki struggled to beat defending champion Dominika Cibulkova 6-2, 6-7 (1), 6-1. Wozniacki, another former No. 1 player, won four consecutive games in the first set and later took advantage of an ailing Cibulkova, who was playing with her right hip bandaged. She took a medical timeout while trailing 2-1 in the third set.
"The difference at the end was that I just could keep my highest level up to the end and I managed to win most of the important points, especially at the beginning of the third set," Wozniacki said.
The 11th-ranked Wozniacki, who finished the last two seasons at No. 1 on the WTA rankings, still has a chance to overtake Marion Bartoli of France at No. 10 by the end of the season if she wins titles in Moscow and at the season-ending Tournament of Champions.
In her seventh semifinal this season, Wozniacki will face Sofia Arvidsson of Sweden.
The 46th-ranked Arvidsson, who is looking for her second title this season, broke twice in each set to eliminate the only Russian left at the Kremlin Cup, seventh-seeded Maria Kirilenko, 6-3, 6-3.
"Maria is a tough player," Arvidsson said, "you have to play at your best to beat her."
"I hope I'm lucky enough to make my first ATP final," Jaziri said. "But anyway, I'm already happy with my performance here."