The seventh-seeded Ivanovic, from Serbia, won nine of the final 11 games. She used a strong ground game to overpower Cibulkova, the Slovak player coming off a victory Sunday in the Bank of the West Classic at Stanford.
"As soon as the draw came out I thought it was one of the toughest first-round matches, especially being a seeded player," said Ivanovic, the former top-ranked singles player and 2008 French Open champ. "But I was really happy with my performance. This kind of win obviously means a lot to me."
Top-seeded Victoria Azarenka of Belarus beat Italy's Francesca Schiavone 6-2, 6-3 in the second round. Azarenka was playing her first match since withdrawing from her second-round match at Wimbledon because of knee and hip injuries.
"Just going on the court and competing was so much joy for me," Azarenka said. "I was really playing and happy to be doing something I was training for. . . . I don't think about it (injuries)."
Former world No. 1 Martina Hingis was scheduled to come out of retirement for the second time Wednesday night, teaming with Daniela Hantuchova in a doubles match. It will be Hingis' first WTA Tour match since September 2007. The five-time Grand Slam singles champion is scheduled to play doubles with Hantuchova in four tournaments, including the U.S. Open.
Ivanovic, who fired coach Nigel Sears after her second-round loss at Wimbledon, said she has been working on "being consistent and also working on that confidence part. I think it plays a big role. I felt like my game has been there at times but not my confidence."
Third-seeded Petra Kvitova of the Czech Republic and fourth-seeded Roberta Vinci of Italy had few problems in winning their opening matches after receiving first-round byes.
Kvitova advanced to the quarterfinals with a 6-1, 6-2 victory over British teenager Laura Robson. The victory avenged Kvitova's second-round loss to Robson at the Australian Open in January.
Vinci beat American Bethanie Mattek-Sands 6-4.