Zheng ended the run of Canadian teenager Victoria Mboko 6-3, 6-4.
The Chinese eighth seed has now won nine successive matches at Roland Garros following her run to Olympic gold last year.
“We had some really good points, it was not an easy match,” said Zheng.
“Today was a tough battle against Qinwen,” said 18-year-old Grand Slam newcomer Mboko.
“Lots of things to learn and hopefully come back to my next tournament with some more positives and (having) improved on my game.”
Zheng is through to the fourth round in Paris for the second time and will next Russian 19th seed Liudmila Samsonova who brushed aside Ukraine’s Dayana Yastremska 6-2, 6-3.
Zheng and Samsonova will both be bidding for a first Roland Garros quarter-final berth.
Swiatek came through despite a very close second set against Romania’s world number 60 Jaqueline Cristian, winning 6-2, 7-5 in nearly two hours in sweltering conditions on Court Suzanne Lenglen.
“Before it was 20 degrees when I played my first match but today it was 30 degrees. It is not easy to adjust but I have played in every condition,” said Swiatek.
The 23-year-old Pole awaits the winner of a big-hitting showdown between Elena Rybakina and Jelena Ostapenko, who has beaten Swiatek in all six of their previous meetings.
A former world number one, who has fallen to fifth in the WTA rankings, Swiatek has not won a title since her third consecutive title on the Parisian clay in 2024.
Earlier Sabalenka produced a dominant display in a 6-2, 6-3 win over Serbian left-hander Olga Danilovic.
“Olga is a fighter. I enjoyed our battle,” said Madrid Open winner Sabalenka after beating her 34th-ranked rival in 79 minutes on Court Philippe Chatrier.
The 27-year-old Belarusian next plays 16th-seeded American Amanda Anisimova for a place in the quarter-finals. (AFP)