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Williams, seventh favorite, face the ninth planted Polish Agnieszka Radwanska, who beat the Slovak Dominika Cibulkova by 6-0 and 6-3. Williams defeated Bondarenko in the same round a year ago.
Williams lost her only previous match against Radwanska in Luxembourg in 2006.
The American took less than an hour to bend to Bondarenko, with nine aces and four service breaks.
Williams wants to become champion at Flushing Meadows, where he won his last title in 2000 and 2001, after defeating her sister Serena the second year.
Radwanska needed an hour and a quarter to defeat the Slovak couple of 19 years, the number 20 world ranking.
Radwanska won three WTA titles this season and is the most successful player of Polish history.
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