The 19-year-old Kleybanova (seeded 29) was the highlight of the day, beating former world No. 1 Ana Ivanovic in three sets. The Serbian star, who lost last year's final and was the fifth seeded this year, got off to a bad start but battled back to win the second set in a tiebreaker before Kleybanova's attacking tennis proved too much and the Russian sealed a 7-5, 6-7 (5-7), 6-2 victory.
Zvonareva, seeded 7, enjoyed a straight sets victory over Italy's Sara Errani 6-4, 6-1, while Petrova (10) won the first set 6-1 in 30 minutes before Galina Voskoboeva of Kazakhstan retired with a back injury.
The two Russians will now play each other in the next round.
World's No. 3 Safina (3) easily zipped past Kaia Kanepi of Estonia in straight sets 6-2, 6-2 to meet French rising star Alize Cornet (15) in the fourth round.
Unlike his younger sibling, Safin lost on Friday and leaves 18th-seed Igor Andreev as Russia's last hope for the men's singles crown. The 2005 champion lost to Switzerland's Roger Federer (seeded 2) in straight sets 3-6, 2-6, 6-7 (5-7).
Andreev plays No. 9 seed James Blake of the United States on Saturday, while on the women's side Elena Dementieva (4) faces Australia's unseeded Samantha Stosur and Svetlana Kuznetsova (8) is pitted against Alona Bondarenko (31) for a place in the fourth round.