Second seed Serena Williams, who holds three of the four Grand Slam titles, also advanced to the third round after overpowering Kateryna Bondarenko of Ukraine 6-3, 6-2.
Belgian Clijsters, in her first competitive tournament after a two-year break, staged a second-set fightback to win in 70 minutes.
"The main thing I was happy with today was the first-serve percentage was a lot better than the first match," the 26 year-old told reporters.
"Although I started making a few little mistakes here and there, I stayed aggressive. If you start doubting yourself you start defending a little bit more.
"I'm glad I stayed focused and just stuck to my game plan."
Unseeded Clijsters, who gave birth to her first child during her time away from the circuit, took the first set after racing to a 3-0 lead before world number 20 Schnyder could steady.
The Swiss raised her game in the second but the Belgian broke in the 12th game to set up a clash with Russian sixth seed Svetlana Kuznetsova, the French Open champion.
"I think any match now is a good test," Clijsters said. "[Marion] Bartoli [the player she beat in round one], Patty, they're good players.
AGGRESSIVE TENNIS
"Svetlana is kind of in between. She can play really aggressive tennis and mix it up well. I will have to be on top of my game if I want to give myself a shot at winning."
Serena Williams banged down 13 aces in her 6-3, 6-2 win over Bondarenko to set up a third-round meeting with Austrian Sybille Bammer.
Fourth seed Elena Dementieva saw off Belgian qualifier Yanina Wickmayer 6-3, 6-4 while fifth seed Jelena Jankovic advanced with a 7-6, 6-3 victory over Russian Maria Kirilenko in a match including 13 breaks of serve.
"I didn't serve very well and I got a bit lucky at the end of the first set but I am happy I was able to win," the Serb said. "Hopefully I will get better from here."
Jankovic's next opponent will be ninth seed Victoria Azarenka of Belarus, a 6-4, 6-2 winner over Anna Chakvetadze of Russia.
Italian Flavia Pennetta secured a meeting with third seed Venus Williams by beating Hungary's Agnes Szavay 6-2, 6-4, extending her winning streak to 13 matches.
Seventh seed Vera Zvonareva survived a second-set blip to oust Russian Alisa Kleybanova 6-4, 1-6, 7-5 but former world number one Ana Ivanovic of Serbia was upset 7-6, 7-5 by Hungary's Melinda Czink.