The tournament's No. 24 seed beat Kazakhstan's Galina Voskoboyeva 6-2, 6-4 in an hour and 22 minutes, converting five of 11 break points.
Kleibanova is at a career-high 26 on the WTA rankings, having climbed seven places this year.
The 19-year-old Muscovite will play either No. 12 seed Marion Bartoli of France or Belarusian qualifier Anastasia Yakimova.
Russia's Elena Dementieva, seeded fourth, won an all-Russia encounter later in the day, beating Anastasia Pivovarova 6-0, 3-6, 6-3.
Russia's Maria Kirilenko did not fare so well, falling to 15-seeded Italian Flavia Penetta 4-6, 5-7 in an hour and 47 minutes.
In the men's, Russia's Marat Safin defeated Spain's Oscar Hernandez 6:4, 2:6, 6:1 in their second round tie.
