The 20-year-old Pole comfortably beat Kudermetova 6-1, 6-0 in just 55 minutes.
Up next for the rising Polish tennis star in Indian Wells is either Kazakhstan's Yulia Putintseva or Latvia's Jelena Ostapenko.
Putintseva and Ostapenko are set to face off in the third round on Sunday.
Meanwhile, another Pole Magda Linette dropped out from the tournament after a foot injury.
In September, Świątek reached the round of 16 of the women’s singles at the US Open, where she lost to Olympic champion Belinda Bencic of Switzerland.
Świątek last year won the French Open, claiming her maiden Grand Slam title and becoming the first Pole to lift a Grand Slam singles trophy.
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Source:PAP