Park looked to be in trouble on the second extra hole as she found the rough off the tee and Yoneyama had a short putt to capture the title. Yoneyama missed the near-gimmie at No. 11 and the players halved the 12th to extend the match to another hole.
In the waning light, Park rolled home the 10-footer for birdie at No. 13 to earn her third title in her LPGA Tour career.
The final match was an unexpected pairing as this tournament featured a 32-player field with 16 representatives from the LPGA Tour and 16 representatives from the Japan LPGA Tour. Golfers were seeded by their position on their respective tour's money list with the higher-ranked players meeting the lower-ranked players from the opposing tour in the first round.
Park was the 12th-ranked player from the LPGA Tour, while Yoneyama was the 14th seed from the JLPGA.
Yoneyama, who advanced thanks to victories over Mi Hyun Kim, Lorie Kane, Kelly Robbins and Hee-Won Han, struck first with a win at the par-5 third hole. She gained a 2-up edge when she took the fifth hole but Park rebounded before the turn.
Park, who made it to the finals with wins over Mikino Kubo, Karrie Webb, Yuri Fudoh and Carin Koch, earned a win at the ninth to cut Yoneyama's lead to 1-up. The twosome halved the next three holes before Park won the 13th to draw even with Yoneyama.
The duo halved the last five holes in regulation to force the sudden-death playoff.
In the consolation third-place match, Koch beat Han 1-up.
Final results from the Cisco World Ladies Match Play Championship