Olympic gold medallist Zheng, in action for the first time since September last year, beat 2020 Australian Open winner Kenin 4-6, 6-1, 6-2 to reach the second round where she'll meet Alycia Parks.
Now ranked 26th in the world, Zheng made a statement as she threw down 20 aces on the way to winning her season opener.
"When you come back from injury, it's never totally pain-free," Zheng said.
"But generally I believe my elbow is in a good process because I could feel it.
"I'm in a really good stage, and I can keep going for another two hours serving," she added.
Fifth seed Andreeva kick-started her tournament with a 7-6 (0), 6-1 win against Magda Linette after receiving a bye through the round of 64.
However, there were several high-profile eliminations on the day as third seed Amanda Anisimova retired when trailing in the third set to Karolina Pliskova and former US Open Grand Slam champion Emma Raducanu also had to retire in the deciding frame of her meeting with Camila Osorio.
On Tuesday, top two seeds Iga Swiatek and Elena Rybakina will play their first matches in the round of 32.
Fourth seed Coco Gauff will open her singles campaign after falling in the doubles on Monday playing alongside rising star Victoria Mboko.