"I definitely thought about Venus, and I thought what a good comeback Kirsten had, so I thought, 'I'm not going to get overconfident,'" Williams, who next faces Slovakia's Magdalena Rybarikova, told reporters.
"She's obviously a really good player, so I was just trying to stay focused as well as intense."
Suprise Wimbledon champion and seventh seed Marion Bartoli suffered a blow before the U.S. Open, retiring when trailing 7-6 1-0 to Rybarikova.
Bartoli said she felt pain while serving and hitting backhands and thought she had suffered an abdominal injury to her right side.
"It took me so much energy to win my first grand slam that at some point I will have a kind of low, and it's normal," she said.
It was smooth sailing for most of the seeds, however, with Pole Agnieszka Radwanska and China's Li Na advancing, but tempers frayed during fifth seed Sara Errani's tight 7-5 7-6 defeat of Frenchwoman Alize Cornet.
Italian Errani took exception to her opponent encouraging herself in Spanish rather than French and let Cornet know about it.
"I just said to her, 'why are you saying 'Vamos' and not 'allez?'" Errani said. "Normally she says 'allez,' so I thought, why you say 'Vamos'?"
Defending champion and sixth seed Petra Kvitova also moved ahead with a 6-3 6-3 victory over Carlsbad Open champion Samantha Stosur. She will face Cirstea in the quarter-finals.
(Editing by Ian Ransom)
