The No.7-seeded Stosur was the first to win, surviving a two-hour, 12-minute duel against Peng Shuai, dropping the second set and rallying from a break down in the third to beat the No.32 seed, 63 36 62.
"I'd had success against her in the past, so I knew if I could stay focused on the task at hand I'd have a shot," said Stosur, who had won all four of their previous meetings. "When I got down an early break in the third I knew I had to step it up. I got back on serve and thankfully closed it out afterwards."
Stosur was so focused going into the match she didn't even notice the earthquake in the morning.
"I was warming up for my match and didn't feel the ground or anything, but all of a sudden I felt the stands were moving and the flagpoles were going a little bit crazy," Stosur recalled. "I thought it was just a big gust of wind, not an earthquake - but then we found out later it was an earthquake.
"A little bit of excitement to spice the morning up, I guess!"
Next up for Stosur will be No.24 seed Mona Barthel, who upset No.11 seed Ana Ivanovic, 61 36 60. The match looked headed for a photo finish after Ivanovic grabbed the second set, but Barthel shut her out in the third, despite the former World No.1 having game points in three of the six games.
Stosur and Barthel will be playing against each other for the first time.
Azarenka, the No.1 seed and defending champion, was three games from losing at 63 33 down against No.28 seed Kirsten Flipkens, but won 12 points in a row from there to take the second set, then didn't even lose another game the rest of the way to complete a 36 63 60 victory over the crafty Belgian.
"She really played well," Azarenka said. "She came out firing, playing so freely and going for every possible shot there was. I didn't really adjust well - I didn't take my chances, and I was too blurry in what I had to do. But whatever happened in the first set is not going to happen tomorrow."
Azarenka is now 16-0 on the season. "I'm really happy that I find the ability to turn around my matches no matter what the circumstances are on the court," she said. "That's definitely what excites me the most - the game and the shots, it's so not important when you can turn it around with your mentality and with your physical ability, because shots is just shots. Everybody out there can hit shots.
"To know when to take your opportunities is a learning experience. I'm glad I'm starting to master that."
Next up for Azarenka is Urszula Radwanska, who won an all-unseeded battle against Jamie Hampton, saving a set point down 5-4 in the second set and eventually closing the American out in two, 60 76(4).
"I haven't played Urszula in a long time - I only played her once," Azarenka said. "She's a very talented player and I think she's improved her game a lot, so obviously she's been showing some very good results in the past. I have to come up with my best game. I'm sure she will, too. We'll battle."
Azarenka beat Radwanska in their only previous meeting, but her memory of it is foggy.
"I don't remember much. I think it was quite close two sets. I just remember which court we were on."