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Williams, who will regain the No. 1 ranking if she reached the semifinals, showed no signs of the back or ankle injuries that troubled her at the Australian Open during her 6-2, 6-1 win over Russian qualifier Daria Gavrilova.
"I have it heavily taped. As long as the tape doesn't get loose, it feels really good," the 15-time Grand Slam winner said. "Physically feeling better is so good, and hopefully I can stay on that level."
The third-ranked Sharapova, who won the tournament in her two previous appearances, downed French qualifier Caroline Garcia 6-3, 6-2.
American Sloane Stephens, who defeated Williams at the quarterfinals of the Australian Open, beat Anna Tatishvili 6-2, 6-2. Stephens pulled out of Fed Cup against Italy last week after aggravating an abdominal muscle injury and suffering from a stomach virus.
"I hadn't really been serving since my ab was a little sore," Stephens said. I didn't serve that great today. "Actually last weekend I got like, I don't know, like a stomach virus or something. ... I'm just happy that I can play now."
Williams broke Gavrilova to go up 4-2 in the first set and took a 5-0 lead in the second. Williams then saved five break points before serving out the match when the 160th-ranked Gavrilova hit a forehand long.
Williams
"I wasn't making my first serve today," she said. "It's always something I can work on and try to do better in my next round."
Sharapova broke the 172nd-ranked Garcia twice go up 3-0 in the first set. Garcia won the next two games but Sharapova broke for a third time. The second set went with serve through the first five games until
"I was quite happy with the way I played, because my opponent has already played a few matches here so she's been able to get in that match atmosphere," Sharapova said. "So I wanted to start really strong. I had a tough match against her in my previous encounter where I played three sets. I really wanted to start better this time as opposed to last time."
Sharapova was joined in the third round by 11th-ranked Marion Bartoli, who downed former French Open champion Francesca Schiavone 7-6 (5), 6-3. The loss leaves Schiavone, who has fallen to 54th in the rankings, winless in four matches this year.
Sixth-ranked Angelique Kerber became the first top player to be eliminated from the tournament, routed by fellow German Mona Barthel 6-1, 6-2 in just more than an hour.
Schiavone had the upper hand early, taking a 4-1 lead. But Bartoli turned things around, saving five break points to make it 4-2 breaking back in the next game when Schiavone doubled faulted. She won the tiebreaker with a return winner.
After trading breaks early in the second set, Bartoli took advantage of Schiavone's errors to go up 4-2. She closed out the match when she chased down a volley and fired a backhand winner down the line.
"The first set was extremely tough," Bartoli said. "It was a hard battle. I knew from coming from indoor clay court I wouldn't play my best tennis for
The only other seeded player to lose Tuesday was No. 12 Maria Kirilenko of Russia, who retired with a right shoulder injury against fellow Russian Ekaterina Bychkova after trailing 3-0.
Yanina Wickmayer of Belgium also withdrew from the tournament with a lower back injury, as did Varvara Lepchenko of the United States with an upper respiratory illness.
Lepchenko won two matches for the U.S. team in its 3-2 loss to Italy in the Fed Cup on Sunday.
