Swiatek, who had never lost to a qualifier at a major before, won three of the last 16 games.
The Olympic champion battled her way through qualifiers amid a post-injury career slump and has recaptured her top form in New York, getting the best of an opponent who had led 7-1 in their head-to-head.
"I start the match not so well and after little by little I found my rhythm," said Zheng, whose chronic right elbow injury stunted her career and forced her to undergo surgery last year.
"I just focused point by point and little by little I started to play better tennis."
She next plays either Naomi Osaka or Elena Rybakina who play later Monday (Tuesday AEST).
Zheng appears to do her best work with her back against the wall, recovering from 5-0 in the deciding set of her third round match with Madison Keys this week, and she found the magic again at Arthur Ashe Stadium, saving three set points to the increasing frustration of the 2022 champion Swiatek.
Swiatek took a medical timeout after the third game of the second set, having a physio come and massage her left thigh, but did not appear to struggle as she returned to the court.
Zheng broke her opponent with a backhand winner down the line in the seventh game and broke again on the fourth match point, leaving Swiatek in tears again two years after she lost the same match-up in the Olympic semi-finals.
Fifth seed Mirra Andreeva also fought back to progress, albeit less dramatically. She beat Anastasia Potapova 5-7 6-4 6-3 to extend her deepest run at the US Open.
The 19-year-old was 5-3 up in the third when Potapova went down with a leg injury late in the match, putting a disappointing end to an otherwise competitive affair.
"She's my good friend and seeing what a battle we had... seeing her fall like this, it's not easy," said Andreeva. "It's not the way obviously you want to finish the match... I just told her she's been playing amazing so far."
The Roland Garros winner will next play the winner of an all-American clash between Iva Jovic and Coco Gauff.
In a tightly fought opening set Andreeva made too many errors, including two double faults in the final game.
But the Russian cut down the mistakes in the second set, thriving in the shorter rallies before Potapova handed over the decisive break with a double fault of her own.
Andreeva was on top in the final set when Potapova collapsed onto the court clutching her left knee in pain after chasing a ball along the baseline in the ninth game. She continued after treatment but was in no state to turn around the match.
