The Italian flew through the opening set before losing five games in a row from 3-3 in the second as her 19-year-old opponent turned the match on its head.
But Errani recovered to break back in the decider before sealing victory in a tie-break.
"She [Zheng] played an unbelievable match," Errani told the WTA official website. "Maybe I didn't play my best today, but for sure she played really well from the second set. I'm pretty tired, but I'm happy I don't play tomorrow!"
China’s Zheng Jie fended off 14 break points as she beat Shahar Peer 6-0 6-3 to seal a spot in the next round.
"I got a little conservative in the second set - this was my first match back after off-season training, and I really wanted to get the win - but my opponent also came alive in the second set," said Zheng.
In New Zealand, fifth seed Jamie Hampton fought back to beat wildcard Tamira Paszek 1-6 6-0 6-3 at the ASB Classic while Venus Williams also progressed with a 6-3 7-6 win over Andrea Hlavackova.
Day One Results (Shenzhen Open)
Patricia Mayr-Achleitner bt Risa Ozaki (q) 6-2 6-1
Viktorija Golubic (q) bt Aleksandra Wozniak 4-6 6-0 6-2
Annika Beck (8) bt Ajla Tomljanovic 6-3 6-4
Jana Cepelova bt Lyudmyla Kichenok (q) 4-6 6-3 6-1
Anna-Lena Friedsam (q) bt Bojana Jovanovski 6-1 7-5
Zheng Jie (7) bt Shahar Peer 6-0 6-3
Barbora Zahlavova Strycova bt Olga Govortsova 7-5 6-3
Vania King bt Petra Martic 7-5 4-6 6-3
Sara Errani bt Zheng Saisai (w) 6-0 3-6 7-6(3)