Suarez Navarro scored a 6-4, 6-3 win over Ishizu and advanced to the next round at La Costa.
Other winners included American qualifier Coco Vandeweghe, who routed Allie Kiick 6-1, 6-2, and Sesil Karatantcheva of Kazakhstan, who ousted Julia Goerges 6-3, 7-5.
Urszula Radwanska of Poland battled her way past qualifier Marina Erakovic of New Zealand 7-6 (7-3), 6-4.
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