Fourth seed Garcia lost the first set to Anastasia Potapova before fighting back to claim a 2-6 6-2 6-2 win.
Seven-time grand slam champion Williams could not produce a similar recovery, however, losing 6-3 4-6 6-3 to Rebecca Peterson, who maintained her upward momentum after winning her maiden WTA singles title in Nanchang last month.
Williams spurned two match points in a defeat to Belinda Bencic in the second round of the China Open last time out and was unable to deliver the kind of display that saw her push a top-10 player to the limit as she fell at the first hurdle.
Sixth seed Yulia Putintseva needed three sets to see off Astra Sharma 6-1 1-6 7-5, while Sam Stosur - a finalist in Guangzhou last month - also had to go the distance to beat Yang Zhaoxuan.
There was home success for Wang Yafan, the Chinese dishing out a bagel in a straight-sets defeat of Lauren Davis, while Heather Watson overcame Kateryna Bondarenko.
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