A tap-in birdie at the par-five 18th for a three-over-par 75 lifted her into a three-way playoff with fellow South Korean Lee Jeong-eun6 and England's Bronte Law at Lake Merced.
Kim then birdied the same hole in the playoff, sinking a three-footer after her two rivals had missed from about double that distance.
"It was really frustrating, starting really bad," 25-year-old Kim told Golf Channel.
"Even then I try to keep going and focus."
"My position is last group so all day I've got a lot of pressure. I'm still almost my heart came out."
Law had threatened to pull off a comeback for the ages after starting the day 10 strokes behind Kim.
She eagled the par-five 15th, shot 65, and then waited for a couple of hours to see if her score would be good enough.
It almost was, but first Lee came in with a 67, before Kim matched them both at seven-under 281.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ian Ransom)
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