FSU starter Mike Compton (12-2) allowed two runs and six hits in six innings. Brandon McNitt (8-4) went 32/3 innings and gave up nine runs, four earned.
"Hopefully this team has done some things that will have a ripple effect that goes on for a long, long time for our athletic department, our university, the Long Island community," said coach Matt Senk.
Stony Brook had overcome big deficits and staved off elimination five times in the NCAA tournament. But the Seawolves couldn't recover from shortstop Cole Peragine's two-run throwing error that opened the door for Florida State's six-run third inning.
"It's a hard loss," third baseman William Carmona said, "but I look back on the season and I think we did what no one thought we could ever do, what everyone thought was basically impossible. We made it happen somehow, and I'll never forget that."
South Carolina stays hot. Erik Payne's bases-loaded triple gave two-time defending champion South Carolina the lead in a five-run fifth inning, and the Gamecocks went on to beat Florida, 7-3, late Saturday night.
South Carolina (46-17) extended its record NCAA-tournament win streak to 22 games by beating its SEC rival and 2011 national runner-up for only the second time in five tries this season.
South Carolina is 27-1 in the NCAA tournament the last three years, the best three-year tournament winning percentage in history.