Wild card Anisimova, meanwhile, took out ninth seed Petra Kvitova 6-2 6-4, ending the Czech's win streak at 14 matches following tournament titles in St. Petersburg and Doha.
It was another highly impressive performance by junior U.S. Open champion Anisimova, who became the first 16-year-old to reach the fourth round since Viktoriya Kutuzova in 2005.
"I'm still in shock," Anisimova, who is ranked 149th in the world, told reporters.
"She's the best player I have ever played, and it was the biggest court I have ever played on."
Anisimova, born in New Jersey to Russian parents, has not dropped a set in three matches. She defeated 23rd seed Anastasia Pavlyuchenkova in the second round.
"I was enjoying the moment, but at the same time, I was trying not to make myself nervous because of the huge crowd, and everything, and the big stage," she said.
"I have been working on that a lot, just not showing any emotions, just fighting the whole time.
"I have been doing that really well these past two weeks."
French Open champion Jelena Ostapenko, seeded sixth, was beaten 6-3 6-3 by Croat Petra Martic, while French 14th seed Kristina Mladenovic were also sent packing in straight sets, going out 6-1 6-2 at the hands of China's Wang Qiang.
U.S. Open champion Sloane Stephens enjoyed a satisfying second-round win over Victoria Azarenka, who beat the American three straight years at the Australian Open from 2013-15.
"I was a little more aggressive," said Stephens, who won 6-1 7-5.
"These courts suit my game really well. I was super focused on just playing my game and making sure I was executing."
It was the first tournament in eight months for Azarenka, who has been involved in a custody battle over her 14-month-old daughter in California where she lives.
"Obviously a lot of work to do," said the Belarusian.
"For me, the most important is to have a clear head, which obviously right now is not.
"Once I figure all my stuff outside of the court, I'll be able to focus and be 100 percent when I'm on the court to be there."
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford)
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