The Norwegian had entered the final day with a five-shot lead but she shot the worst round of her week by six shots while Matthew ate into her lead to force a play-off.
However, Pettersen kept her composure to sink a five-foot birdie putt on the third play-off hole at number 18 to clinch her ninth career victory.
Yani Tseng finished third at 10 under par and recorded her first top-10 finish in 13 starts.
Pettersen got off to a shaky start with a double bogey at the par-three third and after picking up a shot at the fourth, she then bogeyed the eighth.
Eventually, she posted three birdies, one bogey and two double bogeys for a two-over 74, finishing 11 under par.
Matthew started the day seven shots off the lead and carded six birdies and one bogey to force the play-off.
Although Matthew had the momentum, Pettersen's belief did not waver.
She told the LPGA website: "I felt like I had it all in my hand both the first time around, second time around, and the third around," said Pettersen.
"It's getting closer. I'm getting closer to the hole.
"But when you have a putt I would say inside eight feet, you feel like you have it in your own hand.
"You're not going to get too many of those. You would expect Catriona to hit it closer or kind of have a chance, as well.
"I was fortunate enough, but I put enough pressure on her to kind of give myself a good chance.
"And the third time around I just made a solid stroke, and it dropped."