Schnyder, who won her 11th career title in Bali last month, was upset by Czech teenager Petra Kvitova 6-4, 3-6, 6-4.
Chakvetadze was beaten by Romanian qualifier Monica Niculescu 3-6, 6-3, 6-2, her fifth loss in six matches since reaching the New Haven final in August.
The only other seed in action, No. 5 Agnieszka Radwanska of Poland, defeated Russian qualifier Vera Dushevina 6-4, 6-1.
The 18-year-old Kvitova announced her arrival on the WTA Tour at Memphis in February when, ranked 143rd, she became the lowest-ranked opponent ever to beat Venus Williams.
Schynder became her third win over a top-20 player this year.
In their third set, the Swiss star led by 3-0 and two breaks.
"In the third set I was very bad,'' Kvitova said. "But I hit winner, winner, winner and I was in the match again.''
Kvitova rallied to trail 3-4 then conceded only two more points, earning the victory when Schnyder netted a drive volley.
"It was a strange match,'' Schnyder said. "Her game was in and out and I was not ready for the rallies when they came.''
Niculescu, with her injured left thigh heavily strapped, won 10 of the last 13 games to oust Chakvetadze.
"I can honestly say this is my best career win,'' Niculescu said. "I don't know how I finished because I was very tired. I think I had some cramps.''
Niculescu called for a trainer while leading 2-1 in the second set. The delay appeared to affect Chakvetadze's timing - she was broken in her next two service games and trailed 5-1.
In the third set, Chakvetadze double-faulted to hand Niculescu a 3-2 lead, and then netted a backhand to lose serve again and trail 5-2.
Also, 2007 Wimbledon finalist Marion Bartoli of France defeated Agnes Szavay of Hungary 6-4, 6-0 to set up a second-round match with second-seeded Ana Ivanovic, Anabel Medina Garrigues of Spain will meet Olympic champion Vera Zvonareva after beating Li Na of China 6-2, 6-4, and Francesca Schiavone of Italy beat Caroline Wozniacki of Denmark 7-6 (3), 6-2.
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