Keys converted six of her 10 break points in the match.
"I feel good. I'm always a little bit nervous ahead of my first matches," Keys said. "It's always hard to get out there and get all the nerves out. I started a little slow but I was really happy to get the win."
The American will next face Zarina Diyas of Kazakhstan.
Also, Hsieh Su-wei of Taiwan saved a match point before defeating Garbine Muguruza of Spain 3-6, 7-6 (1), 6-1.
Babos will next play third-seeded Karolina Muchova, who dropped a set before defeating Alison van Uytvanck 6-4, 3-6, 6-2.
In other first round matches, qualifier Ana Bogdan beat sixth-seeded Polona Hercog 6-3, 3-6, 6-1, Anastasia Potapova came from a break down to beat wild-card entry Han Na-Lae 7-6 (4), 6-1, and qualifier Priscilla Hon defeated Hiroshima semifinalist Mihaela Buzarnescu 6-3, 6-4.