Olson, who had maintained a firm grip on the lead for majority of the weekend, missed a six-foot putt for bogey on the 18th to force a playoff.
The 26-year-old signed for a 74 to slip into a four-way tie for second spot alongside fellow Americans Austin Ernst and Mo Martin and Kim Sei-young of South Korea.
"I was aggressive with my putt - a little too aggressive obviously," said Olson, whose previous best finish in an LPGA event was tied for seventh in 2014.
"Obviously it's really -- it's disappointing to finish like that, but I had a great week. I have so many positive things to take away from it.
"I'm very pleased. This is my best finish I believe in a major, so that's always a positive."
Stanford ended her long wait for a major title, 15 years after finishing runner-up at the U.S. Women's Open.
She becomes the lone golfer from the U.S. this year to capture a major title, and the first since Danielle Kang's triumph at the Women's PGA Championship in July 2017.
For her sixth and the biggest victory of her career, Stanford will take home $577,500 in prize money.
It was a roller-coaster ride for Stanford, who went eagle, double-bogey, birdie from the 15th, then saw her birdie chance brush the hole at the 18th.
"You know, I know me very well and I did a bunch of stupid stuff, but deep down I'm a fighter, I'm a grinder, and I've always been that," she said.
"I'm grateful. And so happy for everybody at home, everybody that's all cheered for me and never gave up on me."
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
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