Denmark's Caroline Wozniacki, Azarenka and Russia's Nadia Petrova, ranked 8-10, also received first-round byes.
Sara Errani and Roberta Vinci are due to play doubles against Anastasia Pavlyuchenkova and Nadia Petrova.
Sixth seed Nadia Petrova , of Russia, routed Italian Antonella Serra Zanetti 6-0 6-1.
Germany's Sabine Lisicki and Russia's Svetlana Kuznetsova were also winners, beating Dominika Cibulkova and Nadia Petrova respectively.
Meanwhile Anna Kournikova may have to pull out of Thursday's second round match against fellow Russian Nadia Petrova.
Third-seed Nadia Petrova also advanced to the last eight after beating Ana Ivanovic, of Serbia and Montenegro, 7-5 6-3.
He was joined on the withdrawal list by Thailand's Paradorn Srichaphan and Russian women Svetlana Kuznetsova and Nadia Petrova.
Sharapova will play defending champion Nadia Petrova in an all-Russian decider on Sunday.
Wozniacki, 19, moves on to the last eight where she will play Nadia Petrova, who beat Aleksandra Wozniak 6-2 6-4.
Among the seeded women to lose were No. 6 Angelique Kerber, No. 11 Nadia Petrova and No. 14 Maria Kirilenko.
Australian sixth seed Samantha Stosur , the beaten finalist in 2010, breezed past Russian No. 27 Nadia Petrova 6-3 6-3.
The Williams sisters also progressed, with Madrid champion Serena next facing Russia's Nadia Petrova in the second round on Wednesday.
There was better news for Kleybanova's compatriot Nadia Petrova , who needed 70 minutes to beat home favourite Alicia Molik 6-4 6-1.
Top seed Nadia Petrova then beat Austria's Sybille Bammer 7-6 (7-4) 6-1.
Earlier on Sunday, Kuznetsova was overpowered by fellow Russian Nadia Petrova, who won 6-3 3-6 6-1, having routed Kim Clijsters in the third round.
Russia's Nadia Petrova progressed to the quarter-finals, where she will meet defending champion Elena Dementieva, after her opponent, Caroline Wozniacki, retired with breathing problems.
The Serbian will now take on Russian 14th seed Nadia Petrova, who battled to overcome Safina's conqueror, unseeded Romanian Alexandra Dulgheru, winning 6-0 3-6 6-2.
Henin is also on the comeback trail, inspired by Clijsters' exploits, and returned with a superb straight sets win over second seed Nadia Petrova on Monday.
The 17-year-old Czech set up a chance of revenge for her semi-final defeat in Moscow last week against Nadia Petrova, who beat Samantha Stosur 6-1 6-3.
The three-time champion, who is unbeaten in 12 matches after winning two successive tournaments, will next face either Russian 16th seed Nadia Petrova or Slovakia's Daniela Hantuchova.
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The 25-year-old -- a Wimbledon quarterfinalist last month -- also has a chance to earn a podium place in the women's doubles, reaching the semifinals with Nadia Petrova.
Henin, who put out second seed Nadia Petrova of Russia in her first round match, secured a 6-4 6-3 victory in one hour 17 minutes to reach the quarterfinals.
In other first round action, eighth-seeded Australian Samantha Stosur edged past Argentine Gisela Dulko 7-6 7-5 and 16th seed Nadia Petrova beat Russian compatriot Elena Vesnina 4-6 6-2 6-3.
They were joined in the second round by 13th seed Samantha Stosur of Australia, Russian 19th seed Nadia Petrova and former world No. 1 and 2008 finalist Ana Ivanovic, seeded 20th.
Number 14 seed Flavia Pennetta demolished her fellow Italian Roberta Vinci 6-1 6-1, 19th seed Nadia Petrova was similarly comfortable in her 6-1 6-2 win against Agnes Szavay, and Alexandra Dulgheru eased past Timea Bacsinszky 6-4 6-2.
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