With the Bulldogs trailing 59-57, after leading most of the game, Hammonds connected on a 3-point jumper to put Georgia back on top at 60-59 with just under five minutes to go.
On the Bulldogs' next possession, Hammonds followed with another jumper and then stole the ball. A turnover gave the ball back to the Spartans, who cut the deficit to 62-61 on a layup by Malik Moore.
But then Maten took over, scoring the next seven points to give Georgia some breathing room and a 69-61 lead with 31 seconds left in the game.
Maten was the game's leading scorer with 22 points. He also had a game-high 14 rebounds.
Hammonds and William Jackson II each scored 13 points for Georgia.
Moore and Mike Cunningham led the Spartans with 16 points each. Jure Span added 13 and Deion Holmes had 10 points for Upstate.
Upstate's new coach Kyle Perry fell to 1-2, the Spartans' other loss coming in the season-opener against No. 15 Minnesota.
Georgia moved to 2-0 after beating Bryant in its season-opener on Friday.
The Bulldogs led throughout the entire first half and the early minutes of the seconds, with their biggest lead being 11.
And with the score 42-33 at the 17:27 mark of the second half, it looked like the Bulldogs would go on a run and put the game out of reach.
Instead, the Spartans outscored the Bulldogs 13-3 to take a 46-45 lead when Holmes connected on a 3-pointer with 13:02 left in the game.
The two teams traded hoops over the nine minutes, setting the stage for Hammonds and Maten to take over.