Catcher Aaron Cheesman led the Seminole offense going 3-for-4 with a run scored, while Eddy Martinez-Esteve and Gibbs Chapman each had two RBIs.
Brian Schultz (1-0) picked up his first-career win as he pitched two scoreless innings with two strikeouts and two walks, while Tyler Chambliss returned to earn his first-career in the ninth inning.
"Those guys really need a pat on the back. Those Bulldogs from UNC-Asheville were very impressive," FSU coach Mike Martin said.
"They kept their poise. They really battled. They showed great competitiveness and just never went away. The past UNC-Asheville clubs that we played, we would get four or five runs the first three innings and they would roll over and quit. And this bunch just would not go away."
FSU took a 1-0 lead in the bottom of the third inning on Martinez-Esteve's RBI double down the left-field line.
With two outs, Stephen Drew walked and stole second before Martinez-Esteve's double.
UNCA tied the game at 1-1 in the top of the sixth on Matt Roberts RBI single. Ben Padgett singled and moved to second as David Williams drew a walk. With two runners on, Roberts single off the wall in right-center scored Padgett from second for the tying run.
Cheesman and Sauls then scored as Chapman ripped a two-run single into right field in what turned out to be the game-winning hit. Robinson later scored the third run of the inning on Martinez-Esteve's sacrifice fly to right to make it 4-1.
Martin credited the Bulldogs' pitching staff.
"We swung at pitches today that I haven't seen us swing at against Arizona State and against other clubs. I think we were a little bit anxious," Martin said.
"We basically, as they said in the ol' days, fished as pitches we should have left alone. Gosh, we struck out just way too many times -- 10 times. That's not going to cut it when you have a 24-out game if you play eight innings like we did. We had eight at-bats. That's almost half of our outs being strike outs and that's not going to cut it."
Mark Sauls received a no decision after holding the Bulldogs to one run on four hits with three strikeouts and one walk in five innings of work.
"He just a guy who has to understand he can't throw a pitch, when he's ahead of the count, over the plate where he threw that one pitch," Martin said of Roberts' single off the wall that tied the game.
"He pitched a very good game and then, 0-2 he has the guy right where he wants him, and he throws the fastball right over the plate and the guy hits it off the wall. That's a critical part of the ballgame."
Dave Williams took the loss for UNC Asheville as he allowed three runs on two hits in the final three innings. Williams also struck out four and walked three.
Drew went 1-for-1 on the day with three walks, a stolen base and a run scored.
Florida State will travel to Minneapolis, MN next weekend, March 5-7, to play in the Dairy Queen Classic in the Metrodome hosted by the University of Minnesota. The Seminoles will play Nebraska on Friday at 1:15 p.m.ET, West Virginia on Saturday at 1:05 p.m. ET and Minnesota on Sunday at 4:05 p.m. ET.
Information from FSU sports info was used in this report.