Kirilenko's victory has set up a second-round tie with world number 31 Aleksandra Wozniak of Canada.
In the other semifinal, 2006 winner Maria Sharapova will face fellow Russian Maria Kirilenko.
Their chief economist, Andrei Kirilenko, maintains that high-frequency traders are profiting at the expense of smaller traders.
Kuznetsova, meanwhile, will now play fellow Russian Kirilenko after the 30th seed beat Yvonne Meusburger in straight sets.
Another Russian, 18th seed Maria Kirilenko, suffered a surprise 6-3 6-1 defeat by Iveta Benesova of the Czech Republic.
Azarenka beat Maria Kirilenko of Russia 6-3 6-4 in the battle for bronze.
Among the seeded women to lose were No. 6 Angelique Kerber, No. 11 Nadia Petrova and No. 14 Maria Kirilenko.
Kirilenko, who knocked out Maria Sharapova in the first round, progressed with a 7-5 7-6 (7-4) victory against Italy's Roberta Vinci.
Sharapova will next face 21st-seeded Czech Klara Zakopalova, who upset another Russian when she beat No. 14 Maria Kirilenko 6-2 7-6 (7-4).
Then Maria Kirilenko came back to beat Petra Kvitova in three sets.
Denmark's Caroline Wozniacki eased past Urszula Radwanska 6-1 6-2, while there were also wins for Russia duo Maria Kirilenko and Ekaterina Makarova.
"This time I decided to be a lot more aggressive, " said Kirilenko.
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Elsewhere, sixth seed Anna Chakvetadze set up an enticing third round meeting with fellow Russian Maria Kirilenko by beating Alisa Kleybanova 6-3 6-4.
In the women's side of the tournament, World No.1 Serena Williams cruised into the last eight with 6-3 6-1 win over Maria Kirilenko.
The American, who defeated Timea Bacsinszky of Switzerland in her opening match on Tuesday after a first-round bye, will next face Maria Kirilenko.
Maria Kirilenko, at 13th the highest-ranked player in the first-round tie, won the opening singles rubber 7-6 (7-3) 6-4 against 42-year-old Kimiko Date-Krumm.
She had to retire in the fourth round of the Australian Open in the first set of her clash with another Russian, Maria Kirilenko.
Sharapova's semifinal opponent will be countrywoman Maria Kirilenko, after the 13th seed upset in-form world No. 8 Petra Kvitova with a 4-6 6-4 6-3 win.
Kirilenko will meet third seed Agnieszka Radwanska of Poland in the quarter-finals, while top seed Samantha Stosur of Australia will play Yanina Wickmayer of Belgium.
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Kirilenko, the 27th seed, thrashed Japanese veteran Akkiko Morigami 6-1 6-1.
Russian seeds Svetlana Kuznetsova, Anastasia Pavlyuchenkova and Maria Kirilenko also went through, with the latter to face Canada's Aleksandra Wozniak after eliminating Australia's Jarmila Gajdosova.
Henin will now face China's Zheng Jie who cruised into the last four with a straight sets - 6-1 6-3 - victory over Russia's Maria Kirilenko.
Second seed Caroline Wozniacki came from behind to defeat Bulgarian qualifier Tsvetana Pironkova 3-6 6-3 6-4 and set up a third-round tie with Russia's Maria Kirilenko.
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Peer had beaten Kirilenko in New Zealand at the start of the season but this time the Russian was in control, defeating her opponent 6-4 6-3.
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The teenager had only got to bed at 1am after losing her doubles quarter-final alongside Sally Peers against Maria Kirilenko and Agnieszka Radwanska late on Wednesday evening.
But the 22-year-old shut out the distraction to triumph 6-0 3-6 6-1 in a see-saw match against Russia's Maria Kirilenko -- securing a last four place against Wickmayer.
Russian No. 2 Sharapova will play 20th-seeded compatriot Maria Kirilenko, while the remaining quarterfinal tie pits China's French Open champion Li Na against German No. 18 Angelique Kerber.
Russia's Maria Kirilenko beat eighth-seeded compatriot Elena Bovina 6-3 6-1.
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