The score was tied 47-all with 2:40 left before Florida State took control with a 6-0 run. Solomon Alabi and Singleton both made a pair of free throws, then Kitchen hit a layup with 24 seconds remaining.
It was the 1,000 victory for the Seminoles (4-1).
Florida State went up 45-39 with 6 minutes left. Iona, keyed by Jenkins' 3-pointer, scored eight of the next 10 points to tie it.
Kitchen had five points, including a 3-pointer, as Florida State opened the second half with a 10-0 run to take a 35-24 advantage.
The Seminoles trailed much of the first half but took a 25-24 halftime lead on Deividas Dulky's 3-pointer.