On Sunday, former French Open champion Ivanovic lost to Angelique Kerber of Germany in the final in Linz her first appearance in a final since winning her 11th title in Bali in 2011.
Stosur, who won her second title of the season last Sunday in Osaka, won five consecutive games to win the first set and go 2-0 up in the second.
"I did what I had to do," Stosur said. "I made her play more balls when I had a chance to attack deep, so overall it was a pretty solid match."
Stosur rallied to beat Ivanovic in the semifinals last year before falling to Caroline Wozniacki in the final.
"It was a very tough match. She plays differently than lots of other girls. She has lot of kick, a lot of power," Ivanovic said of that match and Stosur. "This year we are meeting a little bit earlier and I really hope I can avenge that loss."
Gasquet, who suffered an early exit at Shanghai last week and dropped to 10th in the rankings, hampering his chances of qualifying for the ATP Finals in London next month, was pushed hard by the unheralded Russian. Gasquet broke twice in the first set, but Donskoy won the second with a break midway through. Gasquet won five consecutive games in the decider and closed the match on his second match point.
"I wasn't playing good enough in the second set," Gasquet said. "But I have experience. I just added power in the beginning of the third and I managed to break him for a 3-0 lead and that made the difference. I began to play with confidence. It was a good win."
Gasquet is chasing fellow Frenchman Jo-Wilfried Tsonga for the eighth and final spot in London.
Earlier, veteran Ivo Karlovic landed 16 aces to eliminate fifth-seeded Denis Istomin of Uzbekistan 6-3, 6-7 (10), 7-6 (6), while qualifier Mikhail Kukushkin of Kazakhstan defeated fourth-seeded Alexandr Dolgopolov of Ukraine 6-4, 6-4.