Ariya frittered away a seven-shot lead with nine holes remaining before parring the fourth playoff hole to edge South Korea's Kim Hyo-joo and become the first player from Thailand to lift the trophy.
Jutanugarn now holds a four-shot lead over Hyo Joo Kim. Somehow, she kept her cool.
Speaking after her win, Jutanugarn said: "I'm really honoured to join the list of players before me".
"Just keep thinking about those two 3-footers I missed", said DeChambeau, who was at 14-under 202.
She looked as if she might blow a seven-shot lead over the back nine, the way Palmer did losing to Billy Casper at the 1966 U.S. Open at The Olympic Club.
Jutanugarn bogeyed the final two holes, sending the championship to a playoff. The format then switched to sudden death, alternating between the same holes.
Ko pulled that shot back at the 16th, but that was the only shot gained during her round as another bogey at the 18th meant she made the turn in one-over par 37. A bogey at the par-five 17th and the lead was again one.
Kim splashed out to 15 feet from long range, but world No 5 Jutanugarn played arguably the best bunker shot of her career as she got her ball to within two feet of the flag.
Ko will, though, start round three no less than 14 shots off the lead in a tie for 52nd. The third round begins with groups of three, starting on the first and 10th tees.
While Jutanugarn was fading, the 22-year-old Kim was rolling.
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It was a "strange" feeling for someone so unaccustomed to leading, and she could not maintain the torrid pace coming home, playing her final nine holes in even par.
Ko, who only just scraped into the second half of the tournament right on the cut line, finished at 9 over par. She shot a 71 on Saturday and was in a tie for fifth place, eight shots back. She was the second Korean golfer to lose out in a playoff on Sunday after An Byeong-hun lost to Bryson DeChambeau at the PGA Tour's Memorial Tournament in Dublin, Ohio. A birdie at the first extra hole, the 14th, gave her a quick advantage. They would need extra holes to decide who would win their second Major Championship, and who would lament missed opportunities.
Kyle Stanley ran off four straight birdies toward the end of his round until his final tee shot caromed off a tree and across the fairway into deep rough.
Smith works with coach Sean Foley formerly of Tiger Woods fame and Smith is aware of the impact he has as a technical coach and motivator. I think I rushed putting them in. He closed with a 72 and ended in a six-way tie for 23rd.
It was the second of two weather disruptions throughout day two the field to return at 6.45am on Saturday local time to complete their rounds.
Lee, meanwhile, slipped down the leaderboard with a second-round 75 that left her eight adrift.
Jutanugarn, however, hit a great bunker shot.
Olesen held his nerve, however, and got up and down.
"I told myself to just go out and have fun and be really committed, and I did", Ariya said. I was sort of holing a lot of putts. That's why this one feels so wonderful, " Olesen added.