Lewis had three chances to win on the 18th. She missed birdie putts from 13 feet in regulation, 15 feet on the first extra hole and 12 feet on the second.
Playing a group ahead of Lewis, Lincicome made a 10-foot eagle putt to match Lewis at 9-under 279. The long-hitting Lincicome, also the 2009 winner, finished with a 69.
Lewis, the 2011 champion, closed with a 70.
Lincicome lost a playoff to Inbee Park last year in the LPGA Championship, one of the tour's five majors. The 29-year-old Florida player, a six-time LPGA winner, is projected to jump from 18th to sixth in the world ranking.
Two strokes ahead of playing partner Sei Young Kim and four ahead of Lincicome with five holes left, Lewis missed a chance for her third major title. She has 11 LPGA victories and 18 runner-up finishes, three in seven starts this year.
The 30-year-old Texan bogeyed the par-4 13th and 15th – missing from 6 feet on 13 and 5 feet on 16 –and closed with three pars.
Morgan Pressel was third at 8 under. She nearly holed her approach on 18 for eagle and made a short birdie putt for a 70.
Kim, three strokes ahead of Lewis entering the round, had a 75 to tie for fourth at 7 under with Carlota Ciganda (68) and Anna Nordqvist (69). Kim had only one par on the back nine, on the 18th when she needed an eagle to get into the playoff.
Lexi Thompson, the winner last year, was 6 under. She finished birdie-eagle for a 70.