Ko was the youngest woman to win a major when she won the Evian Championship at the age of 18 in 2015.
She followed that with victory in what is now the Chevron Championship in 2016 but, despite having 11 top-10 finishes in the majors since then, she had been unable to add to her tally.
And her hopes of doing so on Sunday were in the balance during an intriguing finale with rain and wind battering the Old Course, as it has done all weekend.
Ko started three shots behind Shin but birdied the fourth, 10th and 14th holes to put herself firmly in contention.
However, Korda looked to have wrestled control with a run of three birdies in four holes around the turn, and she led by two on eight under on the 14th tee.
But a couple of poor wedge shots around the green led to the American taking seven shots on the par-five hole.
“It’s golf,” said Korda. “Unfortunately I messed up over the weekend twice coming down the stretch. That’s what cost me the tournament but I played well.”
Meanwhile, Ko, who was playing in a couple of groups ahead, made a terrific up and down for par from the back of 16 to remain at six under.
That made it a four-way tie for the lead with Shin and Vu, who birdied the 14th.
As the rain intensified Ko hit arguably the shot of the championship, knocking her second to the notoriously tough 17th to 15 feet and two-putting for par.
Her birdie on the last set the target at seven under and she watched from the practice putting green as first Korda, who hit her second shot into the Road Hole bunker, and then Shin, bogeyed the 17th to effectively end their hopes.
That left Vu.
The American world number two holed a 10-footer for par on 17 to keep her in the game.
But she was unable to match Ko’s three up the last and then lost concentration to dribble her par putt wide as she finished with a scruffy bogey.
When asked where she would rank this victory alongside her Olympic title and two other majors, Ko replied: “That’s like asking me who I like best, my mum or dad.”
England’s Charley Hull, who led after round one on five under, started Sunday five shots off the pace but her challenge was sunk by successive bogeys on the sixth and seventh, and a double bogey on the ninth.
Hull, ranked 10th in the world, closed with a three-over 75 and one over total.