Anisimova moves on to play No. 16 seed Elise Mertens. Davis will face 20th-seeded Garbine Muguruza. Canadian teenager Bianca Andreescu, a semifinalist last week in the Mexico Open, got by Irina-Camelia Begu 6-7 (3), 6-3, 6-3. Also advancing was Stefanie Voegele, who beat American Sachia Vickery 7-5, 6-2. Voegele next plays No. 4 Sloane Stephens.
In women's doubles, second-seeded Timea Babos and Kristina Mladenovic were knocked out in the opening round by Mertens and Aryna Sabalenka, 6-1, 6-7 (4), 10-8. Fourth-seeded Nicole Melichar and Kveta Peschke also lost.
The men begin play in the Southern California desert on Thursday.
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