The Chinese teenager took the opening set 7-6 (7-5) storming back from 5-2, winning five consecutive points to seize the tie-break.
But this seemed to galvanise the Pole who won a bagel 6-0 set to send the match to a decider.
There her experience and quality showed as she clinched it 6-2 to set up a quarter-final with American Jessica Pegula.
She beat Irina-Camelia Begu 4-6, 6-2, 6-3.
Świątek has now gone an unprecedented 32 matches unbeaten with titles in Qatar, Indian Wells, Miami Open, Stuttgart and the Italian Open.
Inspiration from Dua Lipa
After her win against Zheng, she explained how she was able to turn the match around.
“It wasn’t easy to find solutions and to find other tactics and to do something differently, because I wasn’t sure what I was doing wrong,” said the Pole.
“She was playing really fast balls, and it wasn’t easy to loosen up, because I felt a little bit tense.
“In the second set I wanted to focus more and not talk to my box so much.
“I sped up my forehand. Maybe that was the solution,” said the world number one.
“I felt my mind become clearer.
“In the first set I realized I was focusing on the technical stuff and I got more tense, and couldn’t prepare my shots.”
“I started singing songs in my mind. It’s not the first time. It was Dua Lipa, a guilty pleasure.”
What the pundits said
Eurosport experts Tim Henman and Mats Wilander saw it as the first real test of the tournament for Świątek.
“She looked a little frustrated at times. There was some dialogue with the box, which was unusual, but she really knuckled down, got the second set and maintained the momentum,” said Henman.
“Having won so many matches, it’s good to have frustration to give her something to work on to build for the next round.”
Wilander argued that the situation presented unknowns that initially threw the world number one.
“I’m guessing because she hadn’t played her before so there wasn’t a lot of information apart from video analysis. It was more to do with her than her opponent, “ said the seven-time Grand Slam champion.
“She needed encouragement, staying in the moment, meditation and breathing to come through.”
