On a day of epic three-set battles, French Open champion Barty needed more than two hours to tame her Estonian rival, rifling a forehand winner on match point.
Barty set up a meeting with Maria Sakkari, who needed even longer to dispatch Aryna Sabalenka 6-7(4) 6-4 6-4 on a hot and humid day.
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Anett Kontaveit of Estonia celebrates a point against Ashleigh Barty

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The Hawk-Eye review system played a key roll in Barty's win when the Australian, trailing 5-4 and 30-15 in the third set, challenged a serve from Kontaveit that had been called in.
The system showed the ball was slightly long, denying Kontaveit what would have been two match points. Barty went on to win the next three games to seal the victory.
An in form Sakkari battled back to dispatch ninth seed Sabalenka on the Ohio hardcourts.
Sakkari saw five set points slip away in the first set but the Greek continued to fight and got help from Sabalenka, who double faulted 17 times and needed a trainer apply ice to her right ankle late in the match.

Maria Sakkari

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Venus Williams also went the distance before claiming a 2-6 6-3 6-3 victory over Croatia's Donna Vekic.
The 39-year-old Williams has struggled at times this season but has been resilient in Cincinnati, defeating defending champion Kiki Bertens in three sets in the previous round.
"There are no easy matches out here," Williams said.
"I cannot emphasise that enough."
The seven-times Grand Slam champion will have her work cut out in the next round when she faces either Wimbledon champion Simona Halep or big-serving American Madison Keys.
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