Clijsters who retired in May 2007 and gave birth to a daughter in February 2008, knocked 12th-seeded Marion Bartoli in straight sets on Monday. The 2005 U.S. Open champion followed up by never losing her serve against the 20th-ranked Schnyder.
"Does it look easy?" Clijsters said. "I started off playing well. It's a really good feeling. I'm moving well. I'm very happy with the way I'm starting. It helps your confidence to get into a rhythm."
The 26-year-old Clijsters is scheduled to meet sixth-seeded Svetlana Kuznetsova in the next round, with a possible quarterfinal match looming against top-seeded Dinara Safina. Clijsters has won six of her seven matches against Kuznetsova.
"I knew coming in that this was a big tournament," Clijsters said. "I knew it was going to be tough from the start, and I'm glad I'm still going. Patty's a good player. We've had a lot of tough matches, a lot of three-setters, so I'm glad I got out in two."
Seeded players swept almost all of the early second-round matches at the Western and Southern Financial Group event, with only No. 7 seed Vera Zvonareva dropping a set. She beat Alisa Kleybanova 6-4, 1-6, 7-5, while ninth-seeded Victoria Azarenka moved on with a 6-4, 6-2 win over Anna Chakvetzade and 14th-seeded Flavia Pennetta scored a 6-2, 6-4 win over Agnes Szavay.
Fifth-seeded Jelena Jankovic, playing her first match after first-round bye, advanced with a 7-6 (6), 6-3 win over Maria Kirilenko.
"First match for me, it's never easy," Jankovic said. "It takes time to get used to the conditions - to playing matches again."
Also advancing were Daniela Hantuchova with a 7-5, 1-6, 6-4 win over Alona Bondarenko and Peng Shuai with a 5-7, 6-2, 6-1 win over Maria Jose Martinez Sanchez.
Williams sisters, Clijsters lead U.S. Open fieldEto’o closer to Inter
