Duncan, who captured the Western Amateur and the North and South Amateur in June, became the first medalist to win the Women's Amateur since Amy Fruhwirth in 1991.
'I don't know what to say,' said Duncan. 'I'm flabbergasted. [Perrot] made some really clutch putts.'
Perrot, the winner of the U.S. Girls' Junior Championship last week in Mission Hills, Kansas, was looking to become the first player to claim the junior title and the U.S. Women's Amateur in the same year.
The 17-year-old from Santiago nearly turned the trick after battling from five-down after 20 holes of the scheduled 36-hole final, which was played in front of a gallery of 3,000 in 100-degree heat.
Perrot evened the match with a par to Duncan's three-putt bogey at the 34th hole -- the par-4 16th. The two halved the 17th and moved to the par-5 18th all square.
After Perrot knocked her third shot to 25 feet at No. 18, Duncan applied the pressure with an approach that settled six feet from the hole. A confident Perrot rolled in her birdie putt to turn the tables, but Duncan responded by making her birdie try to force extra holes.
The playoff started and ended at the par-3 10th. Perrot's tee shot left her with another 20-foot-plus putt for birdie; Duncan struck a 7-iron that landed just two feet from the cup.
Moments after Perrot's bid for birdie came up a little short, Duncan sank her knee-knocker for the win.
'It was a tap-in, but at the same time, it was a two-footer to win the U.S. Amateur. Wasn't exactly a tap-in,' Duncan said.
Duncan was brilliant over the morning 18, firing a 65 to jump out to a 4-up lead over Perrot. When Perrot bogeyed the 20th hole -- the par-4 2nd -- Duncan's advantage was increased to 5-up with 16 to play.
Perrot trimmed her deficit to four holes with a birdie at the third. Although her birdies at the fifth and sixth were matched by Duncan, Perrot closed the gap to two with unanswered birdies at the 10th and 11th.
Duncan lead was only 1-up after she double-bogeyed the 13th hole, the 31st of the match.
In Friday's semifinal matches, Duncan beat Emily Bastel of Upper Sandusky, Ohio, 3-and-1, and Perrot notched a 3-and-2 victory over South Korea's Joo-Mi Kim.
On Tuesday, Duncan finished 36 holes of stroke play tied for first with Celeste Troche of Paraguay to earn co-medalist honors. Troche was defeated by Angela Jerman in Wednesday's second round of match play.