The 28-year-old American saved four break points in the match, one in the first set and three in the second. Bondarenko, ranked 33rd, saved two match points while serving at 5-3, but sent a forehand long on match point in the next game.
"I played OK and when the points became important I was winning those, especially on my serve,'' Williams said. "I was trying to put more into coming in (to the net) on different kinds of shots. I'll be looking to improve my level with every round.''
Williams will next play Francesca Schiavone of Italy, who beat Maria Kirilenko of Russia 6-3, 7-5.
Also, 17th-ranked Flavia Pennetta of Italy beat Nadia Petrova of Russia 6-1, 6-1, Swedish qualifier Sofia Arvidsson overcame Sybille Bammer of Austria 0-6, 7-5, 6-4, and Katarina Srebrotnik of Slovenia defeated Swiss wild card Timea Bacsinszky 6-3, 6-3.
Pennetta landed fewer than half of her first serves, but still never faced a break point. She broke Petrova's serve five times.
The Italian next faces top-ranked Jelena Jankovic.
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