Earlier this year in the Sydney final, Radwanska handed the Slovakian a humiliating 6-0 6-0 loss which Cibulkova said affected her for weeks.
But in Sunday's final, the 24-year-old Cibulkova immediately shook off her nerves by winning the first game and fought tooth and nail to seal the win in two-and-a-half hours.
"The difference between Sydney and today was I made the first game and after that I looked at my coach and said 'Here we go, I am here and it's going to be good today,'" the Slovakian said.
"It was big deal for me because I never beat Aga before and she's a really tough competitor and I had to earn every point. It was really tough physically and mentally. That's why I am so happy that I won."
World number four Radwanska entered the final with a 4-0 record against Cibulkova, and appeared to be in control after winning the first set with creative and steady play.
Cibulkova then settled down, breaking Radwanska to lead 4-3 and holding on to win the second set when her opponent missed a return.
The Slovakian wobbled in the decider, with a double-fault conceding a break and allowing Radwanska to take a 4-2 lead, but she broke back immediately and closed out the contest with a searing backhand crosscourt winner.
She fell to her back in joy and her father Milan jumped on to the court to embrace her.
Cibulkova has flirted with the top 10, reaching a career-high 12 back in 2009, but has struggled to break through to the next level.
She has failed to make an impression at the grand slams this year, but will head into the fourth and final slam, the U.S. Open, with renewed belief.
"I believe I can reach the top 10, but every time I get close I feel so much pressure and I feel these expectations, which is sometimes too much," said Cibulkova, whose win will push her to 21st in the world rankings.
"Maybe (I'll make it) when I get enough experience, and it could be this year, or next year."
Radwanska, who was coming off a disappointing defeat in the Wimbledon semi-finals to Sabine Lisicki, was disappointed with her performance.
"Pretty much everything was a problem for me today," she said. "I didn't play my best and my serve wasn't good. I couldn't hit a ball the whole tournament and I didn't feel like I had good touch here."
(Editing by Ian Ransom)
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