Brissett connected with Matt Dayes on a 35-yard touchdown pass down the right sideline for the go-ahead score with 1:37 left.
Georgia Southern had one last chance, but Kevin Ellison's fourth-down pass from his own 28 fell incomplete with 1:02 left.
Bra'Lon Cherry caught two touchdown passes for N.C. State, which trailed 17-3 at halftime and 20-10 in the fourth quarter.
Ellison passed for 184 yards and rushed for 116 more to lead the Eagles. Matt Breida had a 61-yard touchdown run for Georgia Southern.
It was the first game as a member of the Football Bowl Subdivision for the Eagles, who won six national championships in the Football Championship Subdivision before joining the Sun Belt Conference.
N.C. State, which lost its final eight games of the 2013 season, was in big trouble for much of the game.
Ahead 20-10, the Eagles stopped N.C. State's Shadrach Thornton for no gain on fourth and 1 from the Georgia Southern 9 on the first play of the fourth quarter. They took the ball and were driving when N.C. State came up with a momentum-changing turnover.
On second and goal from the 1, Ellison never got the ball on the center-quarterback exchange. N.C. State's Monty Nelson recovered the fumble.
The Wolfpack marched 99 yards on 12 plays, with Brissett finding Cherry for a 13-yard touchdown to cut Georgia Southern's lead to 20-17 with 6:55 remaining.
The Eagles moved back down the field thanks to a 52-yard run by Ellison, but they settled for a 31-yard field goal by Alex Hanks to take a 23-17 lead with 3:35 to play.