The 19-year-old Kono, from Honolulu, edged 14-year-old Annie Park of Levittown, N.Y., by one stroke. Park shot a 69 to finish at 5-under 139 for the two rounds of stroke play.
The top 64 players advanced to match play, which starts Wednesday. The 36-hole championship match is scheduled for Saturday.
Whitney Neuhauser, of Charlottesville, Va., finished third in stroke play at 140.
First-round leader Lizette Salas, of Azusa, Calif., qualified for match play by shooting a 74 to finish at 142.