Lincicome hit her approach shot to within 3 feet of the pin and rolled in her putt to finish with a 3-under-par 69, one shot ahead of Cristie Kerr and Kristy McPherson.
"That absolutely was the best shot of my life," the 23-year-old American golfer said.
Lincicome, who has won three times on the LPGA Tour, had an overall score of 9-under 279.
Kerr (71) and McPherson (72) played with Lincicome in the final threesome.
Lincicome's eagle was her first in the tournament.
"It happened at the perfect time and moment," she said.
Lindsey Wright (70) was fourth at 282 and Suzann Pettersen (66) and Meaghan Francella (69) tied for fifth at 283.
Costly malfunction causes beer flood at Boston-area brewery
Megyn Kelly: Santa Claus and Jesus are both white men