On a thrilling day of action, Petrova was the first through, coming back from 6-3 4-1 down to secure a 3-6 7-5 6-3 victory over sixth seed Sara Errani.
It is the world number 18`s biggest win in the last three years and as a reward she will come up against Stosur, who made light of her 1-10 record against Maria Sharapova, beating the second seed 6-4 7-6 (12/10).
"I lost nine points in a row from 5-3 up in the second set, but I knew I had to turn it around and not let that continue," Stosur said after the match.
"I had a good start in the tie-break and it really helped me forget about all of that.
Kerber did not have to pick up a racket to go through, with the top-seeded Victoria Azarenka pulling out of their clash with dizziness.
Next opponent Radwanska did have to play, though, seeing off Caroline Wozniacki 6-4 6-3.