Lincicome, who won the event in 2009, beat Lewis with a two-putt par on the third playoff hole to take home the first women's major of the year.
The 29-year-old managed to set up a playoff with Lewis following her fourth round 69, which included two birdies and an crucial eagle on the 18th hole.
Lewis meanwhile had three opportunities to win the tournament, but missed her 13-foot putt on the 18th, 15-foot putt on the first extra hole and 12-footer on the second extra hole.
Overnight leader Sei Young Kim finished fourth after hitting a fourth round 75, which consisted of five birdies, four bogeys and two double bogeys.
Joining Kim in fourth was Spain's Carlota Ciganda and Sweden's Anna Nordqvist, who recorded a 68 and 69 respectively.
Morgan Pressel claimed third place after carding four birdies and two bogeys during her fourth round 70.
Meanwhile, defending champion Lexi Thompson finished three strokes behind Lincicome and Lewis as she nailed an eagle, two birdies and two bogeys during her round of 70.