Stanford recovered from a gut-wrenching double bogey of her own at the 71st hole with a closing birdie at the last to set the clubhouse lead at 12 under.
But it was Olson - chasing her first LPGA Tour title - who appeared the likely victor, coming to the par-four 18th tee at 13 under and needing only a par to win.
But the 26-year-old pulled her tee shot into trouble, found the green with her third shot and three-putted from long range to fall into a share of second with Mo Martin, Sei Young Kim and Austin Ernst.
Stanford, 40, has been a member of the LPGA Tour for 17 years and her victory in France is her 6th title on the LPGA Tour.
For Stanford, however, whose previous best finish in a major championship came 15 years ago when runner-up at the 2003 US Women’s Open, it was a moment she wondered would ever happen.
“For the longest time I thought I was a major winner,” said Stanford when asked how she felt about the breakthrough. “I thought I was good enough. Not getting it, the doubt starts creeping in, to say the least. I need to process it. Can I get back to you?
“You know, I remember my first time being in contention was in 2003 at the Open. I was in a playoff. I didn't know at the time how close I was because it was only my third year and I had no idea what I was doing, to be perfectly honest.
“As the years go on and you have all the near misses you think, wow, am I ever going to get that close again? I had that moment on 16 tee today. Okay, you know, here you are again. This is as close as you've been in I don't know how long. So now what? We saw what happened.”
Australia’s Katherine Kirk continued to display her liking for the layout for the event having finished 3rd in 2017.
Kirk finished in a share of 10th and was joint-leading Australasian with New Zealander Lydia Ko who produced one of the best weekends with consecutive rounds of 67.
Minjee Lee continued her improvement from a slow start to the week to finish tied for 16th.
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