They both had a match point and Serena eventually won it 7-6 in the third.
Serena had every opportunity to gloat after Wimbledon trophies in both singles and doubles play.
In Monday's other second-round clashes, Serena's fourth-seeded older sister Venus beat Russia's Vera Zvonareva 7-5 6-3.
Her sister Serena, the world number one, will also miss the event in Alabama.
Unfortunately for her, she is due to face Serena Williams in the fourth round.
Serena won the next two points with serves of 107 mph and 108 mph.
Still, Venus was happy with her week, if not how she played against Serena.
This victory by Serena surpassed her 6-2, 6-2 win over Venus at Miami in 2002.
Serena surged to a 4-0 lead and won the first set in 22 minutes.
Australia's Sam Stosur is the fifth seed, with four-time Wimbledon champion Serena Williams seeded sixth.
"Venus is really, really unusually composed and that's just her style, " Serena told CNN.
"She's my bigger sister, she's my older sister, she's a role model for me, " Serena said.
The youngest of five sisters, Serena says she continues to draw inspiration from her closest sibling.
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On the women's side, 30-year-old Serena Williams was stunned by 29-year-old Frenchwoman Virginie Razzano Tuesday.
Before we joined the cruise, the Costa Serena had sailed past its sunken sister.
Serena Williams, who lost in the quarterfinals, will take the number-two rank in the world.
In the women's draw, Serena Williams remains on course for a sixth singles title in Melbourne.
"I definitely knew about it, " Serena told the WTA Tour official Web site www.sonyericssonwtatour.com.
"I can play even better, " said Radwanska, who lost last year's Wimbledon final to Serena Williams.
Serena Williams is already into the second round with a straight sets win over Varvara Lepchenko.
Elsewhere, Serena Williams set up a semifinal showdown with world number one Victoria Azarenka.
Serena Williams continued her march toward the women's final with a 6-1, 6-3 victory over Roberta Vinci.
Serena Williams has been a role model for aspiring young women tennis players for over a decade.
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Sharapova, the reigning French Open champion, lost to Serena Williams in straight sets in the Spanish capital.
For Serena it was her 5th Wimbledon singles title, with the most dominating serve in her sport.
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"I mean she'll never admit it, ever, but I don't think she was 100 percent, " Serena said.
Serena won the final game of the first set at love, and Venus looked out of options.
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