The fourth-seeded Ivanovic, who won her first Grand Slam title at the French Open in June, failed to convert two match points in the 10th game of the deciding set. The Serb has not won straight matches in four events since reaching the third round at Wimbledon in July.
Defending champion Elena Dementieva of Russia beat Katarina Srebotnik of Slovenia 6-3, 4-6, 7-6 (2) to advance to the quarterfinals, along with fifth-seeded Svetlana Kuznetsova of Russia who defeated Italian qualifier Sara Errani 7-6 (0), 6-1.
Serving for the match at 6-5, Cibulkova was broken at love by the fifth-ranked Ivanovic. The 20th-ranked Slovak went 6-4 ahead in a tiebreaker and won on her first match point when Ivanovic returned wide.
"I had a bad first set because I was playing too defensively,'' Cibulkova said. "I got more aggressive, I began to serve better and return better. I knew I had to play more aggressively in the tiebreaker which I did and I'm happy to beat a player of that ranking.''
Dementieva, who won Olympic gold in Beijing, had two breaks to win the first set against Srebotnik. Srebotnik won the next set and saved a break point in the fourth game of the decider before forcing a tiebreaker.
Dementieva won five straight points in the tiebreaker, serving two aces to close the match.
"It took time to adjust to the surface here,'' Dementieva said. "It wasn't easy to win today.''
Srebotnik beat the Russian in straight sets in Tokyo in September.
In first-round action, Danish teenager Caroline Wozniacki beat eighth-seeded Anna Chakvetadze of Russia 2-6, 6-1, 6-4.
After losing the first set and going 1-0 down in the second, Wozniacki won eight straight games against the 12th-ranked Chakvetadze. The 18-year-old Dane, who is looking for her fourth title this season, then broke the Russian at love in the ninth game for a decisive lead.
"It was a tough match. I fought for every point,'' Wozniacki said. "And I came back.''
In the second round, Wozniacki will face Nadia Petrova, who ousted fellow Russian Anna Lapushchenkova 6-2, 6-1.
The winner of the match will play Dementieva in the quarterfinals.
In the men's event of the dual ATP and WTA tournament, Teimuraz Gabashvili upset third-seeded Mikhail Youzhny 7-5, 4-6, 7-6 (1) in an all-Russian match.
Mischa Zverev beat Denis Gremelmayr 6-3, 7-6 (7) in an all-German first-round matchup, and Igor Kunitsyn of Russia ousted Kazakh lucky loser Mikhail Kukushkin 6-1, 4-6, 6-1.
Kukushkin replaced fifth-seeded Dmitry Tursunov, who pulled out with a right shoulder injury sustained during practice on Tuesday.
Maniche Suffers Back Injury
Dementieva advances to Kremlin Cup quarters
