Azarenka recorded a 6-0, 6-1 victory that wrapped up the day's action at the USTA National Tennis Center.
2006 champion Maria Sharapova, defending champ Samantha Stosur and 2011 Wimbledon winner Petra Kvitova also advanced Friday.
Azarenka hit 18 winners to just six for Zheng and brought about a fourth-round meeting with Anna Tatishvili, who eliminated Mandy Minella 7-5, 6-0.
Sharapova, seeded third, stormed past Mallory Burdette 6-1, 6-1. Sharapova, who has played in the finals at three of the last five major championships, will meet Nadia Petrova in the next round. Petrova was a 6-4, 7-5 winner over Lucie Safarova.
Kvitova, playing in the heat of the afternoon, defeated Pauline Parmentier 6-4, 6-4. Marion Bartoli will face Kvitova after her 6-2, 6-4 victory over Kristina Mladenovic.
Stosur, the No. 7 seed after winning the title last year, got past Varvara Lepchenko 7-6 (7-5), 6-2. Britain's Laura Robson battled past Li Na 6-4, 6-7 (5-7), 6-2 to earn a matchup with Stosur.
Robson sent Kim Clijsters into retirement by defeating the three-time U.S. Open champion in the second round Wednesday.
At 18, Robson is the youngest woman among the Top 100 of the world rankings, and this is the first time in her career she has reached the fourth round of a major event.