Denzel Valentine scored 16 points and went six for six at the free-throw line in the final 30 seconds to help
Travis Trice scored 15 points and
had a strong second half for the seventh-seeded Spartans (24-11), who answered a push from the Bulldogs with a poised run to take back control.
Yet they weren't nearly so steady coming down the stretch, flirting with blowing a 10-point lead in the final 1:16. Valentine's free throws were critical as Georgia (21-12) rallied to make it a one-possession game in a finish that at one point had Spartans longtime Coach Tom Izzo turning and punching the scorer's table in fury.
Still, Michigan State did enough to hang on, showing at least glimpses of the experience from playing in 18 straight NCAA tournaments.
Michigan State advanced to face the Belmont-Virginia winner in Sunday's round of 32 in the East Region.
Charles Mann scored 19 points to lead the 10th-seeded Bulldogs, who were playing in the NCAAs for the first time since 2011 and only the second time since 2008. Leading scorer Marcus Thornton had 12 points and 10 rebounds, but battled foul trouble in the second half.
The Spartans led by 13 points at halftime, watched the Bulldogs rally to within a basket, then responded with a 19-8 burst that put them back up by double figures — the kind of steady run expected from Izzo's tournament-tested bunch -- only to be undone by a shaky performance late.
Trailing 64-54, Georgia rallied to within 66-63 on Kenny Gaines' three-point play with 20.9 seconds left. But Valentine hit the third and fourth of his late free throws to make it a two-possession game again, then Mann missed a drive on the next possession that was rebounded by Dawson to set up Valentine's clinching final free throws.
Dawson, who had a quiet first half, had all of his 14 points and six rebounds after halftime, while Bryn Forbes also added 14 points for the game — including a four-point play as Michigan State took a 58-45 lead with 7:16 left.