Zheng, ranked No 7, will face German Jule Niemeier next.
The 21-year-old Chinese defeated American wildcard Amanda Anisimova (4-6, 6-4, 6-2) in the previous round.
In the previous round, Andreeva, ranked No 75, beat Chinese Yue Yuan (6-3, 7-6 (7)).
New York (Grand Slam), other second-round results (USTA Billie Jean King National Tennis Center, hard, USD 75.000.000, most recent results first):
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