Samantha Stosur and Anastasia Pavlyuchenkova also secured spots in the semis. They join Simona Halep, who has already qualified.
Vesnina's victory meant both she and Ivanovic finished round-robin play with 2-1 records, but Ivanovic advanced because of more sets won.
The sixth-seeded Pavlyuchenkova downed Alize Cornet of France, 6-2, 6-2, while Stosur defeated Bulgaria's Tsvetana Pironkova 6-1, 6-4.
In Saturday's semis, Ivanovic will play Halep, and Stosur will face Pavlyuchenkova.