The Croatian claimed success in the third final of her career, overcoming top seed and Australian Open finalist Cibulkova 5-7 7-5 7-6 (7-4) in two hours and 51 minutes.
Vekic threw away a 5-1 advantage in the deciding set after producing a stirring comeback, but bounced back in a tie-break to seal victory at the WTA international hard-court event.
"It was a very tough match," Vekic said. "I was up 5-1 in the third and she just came out playing. I'm so happy I got through in the end."
Slovakian Cibulkova was given a glimpse of the threat carried by Vekic when the talented 17-year-old secured the first break of the match in the fifth game.
The match was then delayed due to inclement weather and that appeared to prove advantageous to Cibulkova as the world number 10 won five of the next seven games to take the set.
Vekic refused to lie down and kept herself in contention during a frenetic second set featuring several breaks of serve.
The world number 95 moved into position to draw level with her fourth break of the match, although she then wasted three set points before finally recovering her composure to hold serve and force a decider.
Vekic's anxiety also played a pivotal role in the final set, but there was no immediate sign of tension from the teenager as she surged into a four-game advantage.
Cibulkova mounted a stunning fightback, though, forcing her way back into the match as Vekic spurned four championship points and dropped her next two service games.
However, Cibulkova's run of five straight games ended when Vekic held serve to set up a tie-break, which she controlled before finally ending an exciting contest with a fierce forehand.