Defending champion Rybakina will next play Katarzyna Kawa, who defeated No.5 seed Belinda Bencic 7-5 6-2 also on Thursday.
The Russian called for the physio at 3-0 down and attempted to continue in the second-round match before opting to stop. She was emotional as she left the court.
"I think Polina, before, she played some tough matches and conditions here are not easy," Rybakina said.
"The balls are quite heavy. I just wish her a speedy recovery and hope it's nothing too serious."
Alex Eala of the Philippines survived six match points before losing 7-5 6-4 to sixth-seeded Diana Shnaider, who reached the quarter-finals in Ningbo for the third time in four appearances.
Meanwhile, in the Japan Open, Katie Boulter breezed into the quarter-finals in less than an hour.
Alex de Minaur's British wife was back on court for the third day in a row after rain in Osaka had disrupted her first-round match but she wasted no time wrapping up a 6-2 6-1 victory over Swiss Viktorija Golubic.
Australian No.2 Talia Gibson, the tournament's fourth seed, will play in her quarter-final in Osaka on Friday against American Ashlyn Krueger.
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