Third-seeded Sam Stosur took advantage of shaky serving from Petra Cetkovska for a 6-3, 6-2 victory in the third round and earned her first consecutive wins of the year.
Fourth-seeded Agnieszka Radwanska saved two set points in the first set and stopped Varvara Lepchenko of the United States 7-5, 6-1. American Christina McHale will play Radwanska in the quarterfinals after defeating Shahar Peer of Israel 4-6, 6-3, 6-4.
Azarenka fell behind 3-1 to Halep, then won eight straight games to lead by a set and 3-0. Azarenka finished the match with another break on a forehand winner down the line.
"I never played her before, so I was adjusting a little bit to her game style. I don't think I really played the right way those first, like, three games, four games," Azarenka said. "But then I started to get my rhythm more and, you know, putting more pressure on."
Two-time Grand Slam winner Kuznetsova lost to Safarova for the second time this year. The 28th-ranked Safarova backed up her win a day earlier over former No. 1 Caroline Wozniacki.
After saving two set points and closing out the first set, Safarova broke Kuznetsova to go up 5-3 in the second.
"I like to play on these courts and I feel confident here. I did very well last night against Caroline and followed that up with a very tough match against Svetlana," Safarova said. "Everything is just clicking together for me this week. I feel very good on serve and it's helping me to win points in key moments."
Radwanska had a much tougher time with Lepchenko. Radwanksa saved two set points and went on to win the next two games to close out the first set. Lepchenko's play became increasingly erratic and she finished with 37 unforced errors compared to 10 for Radwanska.
Stosur, who lost in the first round at the Australian Open and Sydney, used her kick serve to keep the 30th-ranked Cetkovska off balance. She was helped by Cetkovska, who double-faulted three times in the eighth game to trail 5-3. Stosur closed out the first set when Cetkovska's return went long.
Stosur broke Cetkovska three times in the second set, helped by more double-faults from the Czech.