The South Korean, who attends nearby Robina High School on the Gold Coast, become the first amateur to win a major women's professional golf tournament in Australia.
She bogeyed the final hole in regulation to fall into a tie with Cartwright at 13-under 275 on the Royal Pines course.
'I'm happy and excited,' said Yang, who had led after the second and third rounds and had a final-round 70 while Cartwright shot 68.
Yang made a 23-foot birdie putt on the 18th, the first playoff hole, then burst into tears and hugged her father, James, who is also her caddie.
Two other amateurs, Taiwan's Tseng Ya-ni, who shot a final-round 64, and Tiffany Joh of the United States, who finished with a 69, were tied for third at 12-under with Sweden's Louise Stahle (68).
With amateurs taking three of the top four positions, Cartwright picked up the first-place winner's check of $90,000.
Defending champion Karrie Webb, trying for her sixth Masters title, finished tied for 33rd with Britain's three-time Masters winner Laura Davies at 3-under. Davies finished with a 69 and Webb with a 72.
Japan's Ai Miyazato, who led the first three rounds last year before losing to Webb on the final day, finished at 1-over after a 72.
Yang's coach Laurie Montague, who runs a nearby golf academy, said Yang could get some additional invitations to play in international tournaments.
'It would be up to Amy and her parents to explore those opportunities,' Montague said. 'I've never met anyone so humble. Her game is just so solid, she doesn't get too excited. When it comes to maturity on the golf course, she was way beyond her years.'
Cartwright, 22, had mixed feelings about the result. Asked if she was surprised to be beaten by a young amateur, she replied: 'I don't think it matters whether you're 16 or 60 if you have the talent.
'I didn't even know she was an amateur till my caddie told me. I learned a lot about my myself out there today and it will help my confidence.'