Park, the third-round leader, seemed on her way to the victory down the stretch Sunday, but stumbled to three bogeys in the final five regulation holes, allowing Matthew to catch her at 5-under-par 283.
Park finished with a 3-over 75 while Matthew carded a 4-under 68.
After each player parred No. 18 on the first playoff hole and No. 10 on the second, they returned to the 18th, where Park recorded a birdie to earn her third career major victory and second this season.
She also won April's Kraft Nabisco Championship.
The playoff came after a long day of 32 holes -- both the third and fourth rounds were played Sunday due to Thursday's opening round being suspended by rain.