The Norwegian birdied the third extra hole with a five-foot putt to claim the title that she had earlier almost thrown away with a very poor final round 74.
World number one Yani Tseng finished third, one shot back, to record her first top 10 finish in 13 tournaments.
Matthew had started seven shots back, but hit a superb final round 67 that could have been even better had she not bogeyed the 16th to finish at 11-under-par 205.
Pettersen, who had only had one bogey in her first two rounds, had a torrid final day as she made double bogeys on the par-3 third and 12th holes and another bogey on the par-3 8th.
The world number nine admitted that her conservative game plan was working against her.
"After I bogeyed 8, I looked at my caddie, I said, if I am going to bogey, I might as well just fire right at the pin," said Pettersen.
"There's no point playing safe. He said, why don't you just play as aggressive as you have all week. What are you trying to do? So I might have gone out trying to be a little bit protective, which obviously does not fit my game at all."
But she managed to limit the damage with birdies at the 4th, 10th and 14th, then clung on to match Matthew's clubhouse total.
And it was Pettersen - who had led ever since a stunning opening 63 on Friday - who held her nerve as the pair headed up the 18th for the play-off. The duo both parred the hole twice in sudden death before Pettersen made a birdie at the third time of asking against Matthew's par to clinch the title.
"I felt like I had it all in my hand both the first time around, second time around, and the third around," added Pettersen.
"I mean, you would expect Catriona to hit it closer or kind of have a chance, as well (so) I was fortunate enough, but I put enough pressure on her to kind of give myself a good kind of chance. And the third time around I just made a solid stroke, and it dropped."