The world number one and top seed was stretched to the limit by the diminutive Cibulkova before winning 7-6 2-6 6-3 in their first match in the Red Group.
Earlier, Romania's Simona Halep impressed with a straight sets, 6-2 6-4, victory over Madison Keys of the United States in the same group.
Serena Williams is sitting out the eight-woman end of season finale due to injury, making Germany's Kerber the favorite to cap a super season with victory in the $7 million tournament.
But after taking the first set on a tiebreak, by seven points to five, Kerber realized she was in a battle as her seventh-seeded opponent hit back with inspired play.
Three breaks of service saw her level at one set all and when she took an early advantage in the decider, a big upset looked on the cards.
But Kerber, the Australian and US Open champion this season, was made of sterner stuff and immediately retrieved the break before making further inroads against the Cibulkova service.
Forced to serve it out at 5-3, Kerber saved two break back points before closing out victory in two hours 17 minutes.
"It's a challenge for me and I'm trying to play the best tennis in the last tournament in the last week of the year," she said.
"I'm a little bit relaxed now. It was a really good, tough match so I think that was really important for me to have a match like that."
Halep, a finalist two years ago, impressed with a comfortable victory against Keys, who was making her debut in the WTA Finals after a breakthrough season.
She took just 68 minutes to dispatch Keys, winning 13 points in a row to take the opening set in under half an hour.
Keys, who admitted to nerves, put up a stronger challenge in the second set and broke back to level at 4-4 to give herself a glimmer of hope.
It was shortlived, as Halep immediately broke again and served out for an opening victory.
"I panicked a little bit because I missed some easy shots," Halep said. "But I stayed there. I think I did a good job. I was positive until the end."
The White Group, with Agnieszka Radwanska, Karolina Pliskova, Garbine Muguruza and last-minute qualifier Svetlana Kutznetsova, starts Monday.
On the ATP Tour, there was a popular first title in almost three years for Argentina's Juan Martin del Potro, as he took the Stockholm crown.
The former US Open champion, who has battled back from serious injury, beat American Jack Sock 7-5 6-1 in the final Sunday.
He last played a tour final in January 2014, when he beat Bernard Tomic in Sydney.
It is superb preparation for Argentina's Davis Cup final against Croatia next month, where del Potro will lead its bid for a first title in the team event.