Green claimed her third LPGA title, she won the Portland Classic and a major crown at the Women's PGA Championship in 2019.
When her short par putt dropped for the win at the par-three 18th at Wilshire Country Club, Green couldn't hold back her emotions.
"It's been a long few years," the teary 26-year-old from Perth said.
"I played really well last year but getting across the line has been really difficult," added Green, who had nine top-10 finishes last year, including a runner-up finish in the LA Open - which was played on the same Wilshire Country Club course but moved to Palos Verdes Golf Club this year - leaving Wilshire to host this week's inaugural LA Championship.
Green had just two birdies and 16 pars in her two-under 69, rattling in a long birdie putt in regulation at 18 to join the play-off with Lin and Ashok on nine-under par 275.
China's Lin grabbed her share of the 72-hole lead with impressive back-to-back birdies at 17 and 18 to cap a four-under 67 while India's Ashok bounced back from a bogey at 17 to birdie the final hole in her 67.
They returned to the par-three 18th for the play-off, Ashok's 12-foot birdie attempt lipping out as Lin made a 10-footer and Green sank a four-foot birdie to extend the play-off.
Playing 18 one more time, Green was safely on the green as Lin was in the right greenside bunker and was unable to get up and down for par.
Ashok, a four-time winner on the Ladies European Tour, remains in search of a first LPGA title, as does Lin.
Green admitted that she'd had little expectation of lifting the trophy as she tried to keep her patience through 14 pars to start her round.
"I'm really proud of myself for hanging in there because I really didn't think that I'd be in it with how I was playing and making so many pars," she said.
"I knew you didn't need a low score today to win. My caddie just said stay patient. You always try to do that."
Green's cause was helped by some miscues from others, as players struggled to break free atop a crowded leaderboard.
China's Yin Ruoning, who won the LA Open a month ago, was alone atop the leaderboard on 10-under after back-to-back birdies at 14 and 15, but she bogeyed 17 and 18 to finish tied for fourth on 276 with Japan's Ayaka Furue, who carded a six-under 65 to climb the leaderboard.
Overnight leader Cheyenne Knight shook off two early bogeys and was nine-under after her third birdie of the day at 14, but her hopes evaporated with a double-bogey at 15 and her two-over 73 left her tied on 277 with world number one Nelly Korda, who shot a 67, and South Korean Ryu Hae-ran, who carded a 71.
