The current world number one had to overcome her toughest test in weeks as 19-year-old Zheng produced a sensational performance on Court Philippe-Chatrier.
The Chinese teenager saved five set points and came back from 5-2 down in the opening set to force a tiebreak. Zheng then produced another fightback as she came back from 5-2 down in the tiebreak to take the lead.
The opening set lasted a gruelling 82 minutes, and Zheng's incredible effort appeared to leave her gassed for the remainder of the contest.
Swiatek went on to win eight consecutive games and fought off a late Zheng challenge in the third set to complete a 6-7 6-0 6-2 win.
The Pole last lost a set in the semi-final of the Porsche Tennis Grand Prix in Stuttgart against Lidumila Samsonova.
Looking back at her latest win - her 32nd in the ongoing winning streak - Swiatek praised her opponent, saying: "She's playing amazing tennis, I've never had chance to play her before and I was surprised by some of her shots.
"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."
Longest winning streaks in WTA since 2000
Venus Williams - 35
Serena Williams - 34
Justine Henin - 32
Iga Swiatek - 32
Victoria Azarenka - 26
Swiatek's latest victory puts her level with Justine Henin on the list of longest winning streaks in women's tennis since the turn of the century.
The 20-year-old would need to go all the way at Roland Garros in order to go level with current leader Venus Williams.
Next up for the defending French Open champion is a clash with American Jessica Pegula.