The 50th-ranked Cibulkova of Slovakia seized the advantage with a service break in the fourth game of third set for a 3-1 lead and consolidated it with a backhand drop shot to make it 4-1 against the former French Open champion.
A backhand long from Ivanovic, who had her left foot taped midway through the final set, ended the nearly two-hour contest.
"Today I know that my game was there on the court," said the 26-year-old Cibulkova, who cracked the top 10 last year before finishing the season ranked 11th and reached the quarter-finals last week in New Haven.
"I just had to stay tough and know I had to do it, she doesn't give away shots."
Cibulkova, a 2014 Australian Open finalist who earlier this year had Achilles-related surgery on her left foot, will next face American qualifier Jessica Pegula, a 7-5 6-3 winner over Belgian Alison Van Uytvanck.
For Ivanovic, the 2008 French Open champion, the result continued an up-and-down season. The 27-year-old Serb, who reached the semi-finals at Roland Garros this year, was coming off back-to-back quarter-finals at U.S. Open tune-up events in Toronto and Cincinnati.
She also has lost three times to players outside the top-100 this year, including twice at grand slams.
Other early matches on the women's side went according to script as 13th-seeded Russian Ekaterina Makarova dismissed Brazil's Teliana Pereira 6-3 6-3, and 15th seed Agnieszka Radwanska of Poland crushed Czech Katerina Siniakova 6-2 6-3.
(Editing by Frank Pingue)
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