Al Thornton's dunk gave narrowed the gap to one before Bailey drilled a pair of free throws to re-up the lead.
Todd Galloway's desperation three at the buzzer rimmed out to put the improbable win in the books.
"The ACC is one of the best basketball conferences so this win ranks up there," Bailey said.
"We knew they were young and inexperienced and I think that's what it boiled down to."
The inexperienced Seminoles committed 17 turnovers and were outrebounded 33-26 by a much smaller Islanders' team.
Texas A&M - Corpus Christi led almost the entirety of the contest.
An 11-0 at the conclusion of the first half gave FSU a 36-32 advantage that was quickly erased by a 12-1 Islander run after the break.
Back-to-back layups by Galloway and Andrew Wilson tied the game at 65 with 57 second to play before Bailey's heroics.
"This is the first time since I've been here that I've felt that a team we played came with more pressure and more intensity," FSU coach Leonard Hamilton said.
"We didn't appear to be hungry enough to make the little things happen."
"We made some plays but there was never a time that we executed well enough."
Texas A&M - Corpus Christi is a Division I Independent in just its sixth year of competition.
The Islanders beat Texas A&M last season and will test their mettle against Alabama in December. "We didn't have a team six years ago," coach Ronnie Arrow said.
"We had no offices. No basketballs. No pens. No pencils. This means an awful lot to our program and our fans and our school. This win was special."