Williams is playing her first tournament since she regained the No. 1 ranking at the Qatar Open. She lost the Doha final to Victoria Azarenka, but overtook her for the top ranking the week of Feb. 11.
Williams trailed 40-0 in the first and fifth games in the first set, but went on to hold serve both times. In the first game, she uncharacteristically double-faulted on the first and third points.
"I don't think it was too much rust," Williams said. "I think I missed a few shots, but only because I was off footwork-wise. It's always just good to win a match and stay in the tournament and keep going."
Williams is trying to break the tournament record of five titles she shares with Steffi Graf.
"That would be really cool," Williams said. "I have a lot of matches to play. If I can just stay relaxed, I really feel like I can do it, but I'm sometimes my own worst enemy."
Venus Williams had a much tougher road to the third round than her younger sister.
The 18th-ranked Venus Williams eventually prevailed 7-6 (3), 3-6, 6-4 over 78th-ranked Kimiko Date-Krumm of Japan, at 42 the oldest player on the women's tour.
In a match that kept fans cheering for 2 hours, 31 minutes, the 32-year-old Williams finally took advantage of her match point. At ad-out with Date-Krumm serving at 4-5, Williams hit a winning backhand passing shot down the line.
Date-Krumm saved six match points -- one on her own serve in the eighth game, and then five when Williams was serving for the match at 5-3.
"The toughest part is when you reach the finish line and you're not able to get across," Williams said. "Really it's credit to her as she really hit amazing shots. I was pretty determined out there and one (match point) had to come through if I kept putting myself in that position."
Stephens won only four points in her three service games and 10 points overall in the opening set.
"I wasn't playing that great," Stephens said. "I was asleep out there. I definitely just needed to wake up and start playing. A lot just battling and just hanging in there."
"Just very happy to have a quick match," Radwanska said. "The first match is always a little bit tough and tricky."