The winner of 13 Grand Slam tournaments aced her second serve of the match.
"I still have a few weeks 'til (the U.S. Open) and we'll try to do everything to be ready," said Clijsters, who won the tournament in 2005.
Williams got through the first set in 16 minutes using her speed and power, as a bewildered Bondarenko scattered across the court helplessly.
Bondarenko put up some fight early in the second but could do little to contain the American's imposing serve and forehand.
Williams, who is working her way back from a foot injury that put her out of the WTA's top 100, is using the tournament as warm-up for the U.S. Open.
Although she rolled on Tuesday, Williams stopped short of saying she's Grand Slam-ready.
"I would hate to be Grand Slam-ready right now, seeing as how the U.S. Open is not right now, so I'll try to peak at the right time," Williams said. "I want to get better on a few things. I definitely want to try and continue to do more and play harder."
Clijsters shook her head after hitting a forehand and left the match with a reported stomach muscle problem.
It was her seventh appearance at the tournament, having won it in 2005 against Justine Henin.
Despite winning the Australian Open, Clijsters has been hampered this year by injuries. She injured her right ankle in April and then withdrew from tournaments in Rome and Madrid due to right shoulder and wrist injuries.
In other action Tuesday, Roberta Vinci got past Yanina Wickmayer 6-4, 6-2; China's Shuai Peng eliminated Spain's Lourdes Dominguez Lino 6-1, 7-5; Spaniard Anabel Medina Garrigues topped Italy's Sara Errani 7-5, 6-1; Serbia's Bojana Jovanovski won against Jelena Dokic after the Australian retired in the first set; Italy's Flavia Pennetta was a 6-3, 6-1 winner over Russia's Maria Kirilenko; and Slovakia's Daniela Hantuchova beat Argentina's Gisela Dulko 6-3, 6-4.
Dubois, a wild-card entry from Laval, Quebec, fought hard in the first set before taking control in the second. She hit a backhand slice to break Woerle for a 3-0 lead in the second.
Woerle put up resistance when down 4-0, fighting back to deuce several times and causing Dubois to smash her racket against the court in frustration before losing the game.
However, Dubois regained her composure to close out the match and earn a meeting with fourth-seed Victoria Azarenka of Belarus.
Eugenie Bouchard, however, wasn't able to join the Canadians in Round 2.
The 17-year-old wild card from Westmount, Quebec, lost 6-2, 6-2 to 10th-seeded German Andrea Petkovic after a rain delay of almost 3½ hours.