The fourth-seeded Ivanovic led 5-0 in the first set, traded early breaks in the second and rolled the rest of the way.
"It wasn't an easy match for me," Ivanovic said. "I really started well. But she's a tough player and the second set was much tougher that the scoreline indicates. But I added power in the important moments and was very pleased to win."
On Sunday, former French Open champion Ivanovic lost to Angelique Kerber 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 her previous tangle with Stosur. "This year we are meeting a little bit earlier and I really hope I can avenge that loss."
In the men's second round, top seed Richard Gasquet reached the quarterfinals with a 6-3, 3-6, 6-1 victory over Evgeny Donskoy of Russia. Earlier, Ivo Karlovic landed 16 aces to eliminate fifth-seeded Denis Istomin 6-3, 6-7 (10), 7-6 (6), while qualifier Mikhail Kukushkin defeated fourth-seeded Alexandr Dolgopolov 6-4, 6-4.