Another former top-ranked player, Amelie Mauresmo, lost to the in-form Dinara Safina, who has reached her seventh straight quarterfinals. Defending champion Elena Dementieva and Svetlana Kuznetsova also advanced.
However, on the men's side, second-seeded Igor Andreev and third-seeded Mikhail Youzhny were ousted in three-setters.
Ivanovic failed to convert two match points in the 10th game of the deciding set in her first meeting with Cibulkova.
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 the 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.''
Ivanovic won her first Grand Slam title at the French Open in June and climbed to No. 1. But since then the sometime-injured star has lost six of 11 matches and hasn't won two in a row since early June at Wimbledon.
"I created my chances, I served well and I had match points,'' Ivanovic said. "Obviously I'm disappointed.''
Meanwhile, Safina, who lost to Ivanovic in the Roland Garros final, notched another win in her breakthrough season, which includes four titles, runner-up at the Olympics and a career-high No. 3 ranking. She overcame Mauresmo 6-7 (2), 6-4, 6-4.
She served for the first set at 5-4 but was broken by Mauresmo and lost it in a tiebreaker. Both players were erratic in the second set, trading serves three times before the Russian leveled the set score and had an early break in the third set. Mauresmo broke back for 4-5 but double-faulted to go 15-40 and Safina nailed her first match point.
In the quarterfinals, Safina will meet fellow Russian Kuznetsova, who defeated Italian qualifier Sara Errani 7-6 (0), 6-1.
Dementieva beat Katarina Srebotnik of Slovenia 6-3, 4-6, 7-6 (2) and will meet either Russian countrywoman Nadia Petrova or Danish teenager Caroline Wozniacki, who eliminated eighth-seeded Anna Chakvetadze of Russia 2-6, 6-1, 6-4.
Srebotnik beat Dementieva last week in straight sets in Tokyo, but this time the Russian finished off Srebotnik in their tiebreaker with five straight points and two aces.
"It took time to adjust to the surface here,'' said Dementieva, also the Olympic champion. "It wasn't easy to win today.''
Andreev, the 2005 men's champion, served for his match at 6-3, 5-3 but could not put in a single first serve and was broken to love by Jeremy Chardy of France, who went on to win 3-6, 7-5, 6-4.
"I felt that I was losing my serve with every next game,'' Andreev said. "To have such a huge advantage and lose the match - definitely it was my mistake.''
Another Russian, Youzhny, fell to countryman Teimuraz Gabashvili 7-5, 4-6, 7-6 (1).
In other first-round matches, fourth-seeded Paul-Henri Mathieu defeated Sergiy Stakhovsky of Ukraine, 6-3, 6-2, Mischa Zverev beat Denis Gremelmayr 6-3, 7-6 (7) in an all-German 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.
Ivanovic loses to Cibulkova, Dementieva advances
Maniche Suffers Back Injury
