The Belorussian will next play 46th-ranked Czech Iveta Benesova , who won 6-1 6-3 against Russia's Nina Bratchikova.
Of course, because a Belorussian won the Olympic women's 100 meters in 2004, this would be my hope.
The Belorussian, who lost to Serena in the Wimbledon semifinals, will next face Maria Jose Martinez Sanchez of Spain.
Williams hit 26 winners past the Belorussian, taking four of her five break-point chances and denying Azarenka's single break-point opportunity.
CNN: Serena Williams thrashes No. 1 Azarenka in Madrid final
"I love playing Victoria, " said the 14-time grand slam singles winner, who holds an 8-1 career advantage over the Belorussian.
Bate Borisov bubble has now officially burst after Valencia wins 4-2 and the Belorussian side have now slipped to third.
"If I can win that match serving like this, that's pretty remarkable, " said the Belorussian, who will next face 2010 champion Francesca Schiavone.
Williams has won 11 of the 12 matches she has played against Azarenka and triumphed in all five games against the Belorussian in 2012.
Sharapova lost to Azarenka in January's Australian Open final, and the Russian will be seeking to prevent her Belorussian rival from winning a fourth title in 2012.
Doha champion Azarenka and runner-up Williams both pulled out of Dubai due to injuries -- the Belorussian before the tournament, and the American just ahead of her opening match.
Williams said her back also troubled her in last week's Qatar Open, where she was beaten in the final by the Belorussian, who withdrew from the Dubai championships with a foot injury.
Since then Sharapova's career has been hampered by serious shoulder injuries that forced her into a long rehab, but her winning form at Roland Garros will also see the Russian move above Belorussian Victoria Azarenka to take the No. 1 ranking slot on Monday.
"I'm really happy that I find the ability to turn around my matches no matter what the circumstances are on the court, " said the Belorussian, who was criticized after taking a lengthy medical timeout before winning her Australian Open semifinal against Sloane Stephens in late January.
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