Barty fell behind 5-3 in the second set but saved two set points against her American opponent and won four straight games to avoid a third set.
"I had a period in the second set where I let her see far too many second serves, she was able to take advantage of that," said Barty. "In the end, I'm really happy with how I was a bit more aggressive towards the end of the second set and closed it out in straight sets."
The Australian will next face Petra Martic, who beat Veronika Kudermetova of Russia 6-3, 6-1.
Despite hitting fewer winners than the Russian, Svitolina had just 15 unforced errors, compared to her opponent's 33.
"Today I think it was a lot about tactics. Just tried to keep the ball out of the middle, try to work with the ball, try to get the angles and don't rush," said Svitolina
Svitolina plays Alison Riske of the United States in the quarterfinals, who ousted eighth-seeded Qiang Wang 6-2, 6-1.
In the first match of the day, Elena Rybakina advanced after Wimbledon champion Simona Halep withdrew with a lower back injury. Halep was trailing 5-4 in the first set.
Defending champion Aryna Sabalenka beat sixth-seeded Kiki Bertens 6-1, 7-6 (9) to reach the last eight.
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