After a poor six-iron approach into a bunker at the final hole, Green made a clutch up-and-down par, sinking a four-foot putt for the win at Hazeltine National.
She carded an even-par 72 to complete a wire-to-wire triumph, only the second player this century to lead alone after every round.
Green finished at nine-under 279, while last year's champion Park (68) birdied the final hole for second place on 280.
"I was really nervous playing the last five holes," Green said, her voice cracking with emotion.
"To be winning a major as my first event, I'm just over the moon."
The 22-year-old from Perth, Western Australia, came into the championship ranked 114th in the world but now belongs in the same major club as compatriots Karrie Webb (seven) and Jan Stephenson (three).
Webb was the last Australian to claim a major at the 2006 Kraft Nabisco Championship, since rebranded the ANA Inspiration.
Green is a former recipient of the Karrie Webb Scholarship, and seven-times major winner Webb was among a half-dozen Australian friends who mobbed the champion on the 18th green.
FRITTERED AWAY LEAD
Earlier, Green built a handy lead before all but frittering it away with three bogeys in a four-hole stretch around the turn as nerves appeared to take hold.
Her lead was trimmed to one stroke over American Nelly Korda, but Green lifted under pressure.
A 15-foot birdie at the 16th hole steadied Green but Park, playing in the group ahead, was not done.
The Korean birdied the par-four 18th to put the heat on Green and the Australian stumbled with a poor approach shot on the final hole.
Finding a good lie in the bunker, Green nipped the shot nicely but the ball did not stop close enough to take the putt for granted.
After an almost unbearable four-minute wait while the other members of her group putted out, Green rolled in the winning putt and the tears quickly flowed.
Runnerup Park had earlier rued a missed birdie chance at the 17th before making amends at the last.
"There were a lot of putts that didn't go in but I was really glad to make that birdie putt on 18," Park said. "I would like to congratulate Hannah."
Korda (71) and England's Mel Reid (66) tied for third, three shots back.
Thailand's Ariya Jutanugarn, who started the day one shot behind Green, had a nightmare 77 to plunge into a tie for 10th on three-under.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ken Ferris / Ian Ransom)