In the second set, Zvonareva was only able to hold her opening serve. At 2-1, Henin built up three break points but needed only one, taking the game with a cross court forehand winner. She broke twice more to close out the match.
"It was tough for me at the beginning of the match to find a good rhythm. I needed to get used to the conditions," Henin said. "But at 5-3 and then the second set I started to play better."
Henin next will face Agnieszka Radwanska of Poland, who upset seventh-seeded Daniela Hantuchova 6-3, 6-3.
Hantuchova said a cold and nasal congestion contributed to the defeat, which badly damaged her chances of qualifying for the season-ending WTA Championships in Madrid, Spain, Nov. 6-11.
"I had a really hard time today trying to catch my breath in between points," Hantuchova said. "I have been fighting a cold for the past few days and I really just had nothing left and became very tired out there."
It was the first time the pair met on court.
"For sure this is a huge success for me to be in the quarter-finals of a tournament like this," said Radwanska, ranked 31st.
Hantuchova was the fifth of the tournament's eight seeded players to be eliminated.
Wednesday saw the elimination of third-seeded Jelena Jankovic and fifth-seeded Ana Ivanovic, both of Serbia, as well as 2006 Australian Open and Wimbledon champion Amelie Mauresmo, seeded eighth. Sixth-seeded Serena Williams withdrew with a sore thigh on Tuesday.
Besides Henin and Kuznetsova, France's Marion Bartoli -- who was made the ninth seed when No. 4 Maria Sharapova withdrew on Sunday -- is the only other seed remaining in the draw.
Early in Kuznetsova's match, Henin and Bartoli appeared to be the only seeds who would reach the final eight.
Like Henin, Kuznetsova also was playing her first match and struggled to find her rhythm. The Russian was immediately broken at love on her first serve and also lost her second.
The second set was a series of breaks, with Bammer holding only once, while Kuznetsova was able to manage one better.
In the final set, the Russian squandered two break points at 4-3 but made up for it four games later, converting her second match point and sealing the win with a powerful cross court backhand after almost two hours of play.
In the quarter-finals, Kuznetsova will play Italy's Francesca Schiavone, who defeated local favorite Patty Schnyder 2-6, 6-3, 6-4.
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