Lindberg, who had led for all four rounds of the tournament, drained a 25-foot birdie putt on the par-four 10th to set up her victory over seven-time major winner Park at the Mission Hills Country Club in Rancho Mirage, California.
The 31-year-old's birdie left Park needing to make an awkward 10-footer to force an extra playoff hole, but the 2013 champion missed to hand Lindberg the win.
“I just know I'm a grinder. Towards the end yesterday I felt 'This is mine, I'm going to do this',” said Lindberg, who has registered only a handful of top-10 finishes on the LPGA Tour after more than 180 tournaments.
“I just knew I could so I just kept fighting away. I couldn't believe it when that last putt went in,” added Lindberg, whose previous best finish at a major was a share of fifth at the 2015 US Women's Open.
Lindberg celebrated her victory with the traditional dive into the Poppie's Pond hazard at the 18th green — and was joined in the water by her parents, Jan and Gunilla.
“The only reason I'm playing golf is because of them,” Lindberg said.
“It's so great that they were here. I've probably given my dad I don't know how many heart attacks over the past few days.”
“But I'm so happy they're here and I get to celebrate this with them.”
The playoff had extended into Monday after Lindberg and Park finished in near-darkness on Sunday, unable to land a decisive blow through four extra holes.
Their duel resumed at just after 8:00am local time Monday on the par-four 10th, with Lindberg coming within a couple of inches of draining a monster birdie putt that would have completed victory.
Lindberg had to settle for a par however which left Park a 25-foot birdie putt for the championship.
The 29-year-old seven-time major winner's effort rolled up short however, leaving her with a tap-in which she duly converted.
Play then moved to the par-three 17th, but in a sign of mounting tension, both players fluffed their tee shots.
Park's seven iron off the tee kicked into the bunker at the front of the green before Lindberg then squandered an opportunity to capitalise by sending her tee shot into the deep rough.
Both players then chipped onto the green to leave themselves with long putts to make par.
Park ramped up the pressure on Lindberg by draining her putt from 15ft, leaving Lindberg a 12-footer to stay in the match.
The Swede rose to the challenge magnificently, rolling in the putt to take it to a seventh playoff hole at the 531-yard par-five 18th.
After both players found the fairway off the tee, Lindberg then ripped her second shot onto the green to leave herself with a long eagle putt.
Park opted for a conservative approach, laying up and then hitting a perfect wedge to eight feet with her third.
Lindberg steered her eagle attempt wide to around six feet, but Park missed her birdie chance to hand the Swede the chance to drain her putt for victory.
Lindberg's rock-solid putting game deserted her however and she missed to send it back to the 10th tee for the eighth extra hole.
Park appeared to have taken the advantage after hitting her second shot to around 10 feet.
But Lindberg sank her long birdie putt to leave the Korean needing to make hers to save the match, and when her effort missed Lindberg — ranked 95th in the world — had won.