The South Korean rookie birdied the first extra hole moments after a birdie at the 72nd got her into a play-off against Ariya Jutanugarn and Yoo Sun-Young, while Lydia Ko retained the world No 1 ranking ahead of Inbee Park.
Kim birdied three of the first five holes and responded from a bogey at the eighth with further birdies at 10 and 14 before converting a superb approach to five feet at the last to cap a five-under 68 and join Jutanugarn (69) and Yoo (70) on 14 under par.
The top three headed back to the par-five 18th, and Kim's nerveless second found the back edge of the putting surface while Yoo found a greenside bunker and Jutanugarn left herself a tricky pitch from thick rough.
Yoo and Jutanugarn were both unable to get up-and-down, and Kim left her eagle attempt six feet short before holding her nerve to rattle in the winning putt.
"I feel really great," Kim, 22, said afterwards. "I was really nervous. I just focused on my game. I can't find the words. Just crying."
American Brittany Lincicome fired a closing 70 to finish a shot adrift of the leaders, while Park was a further stroke behind after struggling to a 72.
And Park's bid to regain the world No 1 ranking was scuppered when Ko carded an excellent six-birdie 68 to lift herself to 11 under alongside Spanish Solheim Cup star Azahara Munoz and Americans Gerina Piller and Lexi Thompson.
"I wasn't really thinking about it all week," said Ko when asked about the world rankings. "All I could do was try to play my best. I saw the projections on how it would change but I just kept trying to make my putts and make some birdies."

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