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There was some doubt that the seemingly unstoppable world number one was going to extend her winning streak to 32 matches when Zheng snatched the first set despite trailing 5-2.
The 19-year-old from China was also 5-2 down in the tie-break, but reeled off five consecutive points to win it.
However, a thigh injury ultimately cost the unlucky Zheng the chance of a major upset as Swiatek won 6-7 (5) 6-0 6-2 to reach the quarter-finals.
“She’s playing amazing tennis, I’ve never had chance to play her before and I was surprised by some of her shots,” said the 20-year-old from Poland.
“Huge congrats to her. I’m pretty happy I could come back from a frustrating first set and I’m proud of myself that I’m still in the tournament.
“The quarter-final is such a stressful round so hopefully I can loosen up.”
Swiatek will face American 11th seed Jessica Pegula, who beat Irina-Camelia Begu of Romania 4-6 6-2 6-3, in the last eight.
There will definitely be a Russian in the semi-finals with Veronika Kudermetova and Daria Kasatkina to meet each other.