Azarenka took the first set comfortably but faced stiffer resistance in the second from the Ukrainian, who battled hard against the 29-year-old Belarusian on a night when her powerful first serve lacked its usual accuracy.
Azarenka committed 12 double faults and was broken four times but did just enough to edge the defensive-minded Bondarenko, who sent a backhand long on match point to put an end to the marathon match.
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Elsewhere, Australian Ajla Tomljanovic defeated Spaniard Georgina Garcia Perez 6-4 6-1 to set up a second-round meeting with second-seeded American Madison Keys.
Nineteen-year-old Sofia Kenin defeated Paraguayan Veronica Cepede Royg 6-4 6-4 and American Danielle Collins fired three aces and won 76 percent of her first service points to dispatch compatriot Danielle Lao 7-6(2) 6-3.
Greek Maria Sakkari cruised to a comfortable 6-2 6-2 win over American Christina McHale, who committed eight double faults and was broken five times.
Tuesday's action will see 23-times grand slam champion Serena Williams face off against Australian Johanna Konta in their first-round duel in San Jose.
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