Stephon Castle led San Antonio with 25 points, and Devin Vassell added 22 points.
Gilgeous-Alexander emphasized the importance of the win.
"Knowing what it would have meant if we lost this one, we brought the energy from the jump," he said.
Both teams suffered injuries during the game. Thunder guard Jalen Williams left in the first half with a tight left hamstring.
The Spurs lost starting guard Dylan Harper to a right leg injury in the third quarter, compounding their backcourt depth issues as they were already missing All-Star guard De'Aaron Fox due to ankle soreness.
Spurs coach Mitch Johnson acknowledged the challenge of managing a depleted rotation.
"Obviously this team is as good as anybody at turning you over, so when you're down some of your primary creators and initiators it causes a little bit of an extra strain," he said.
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Game 3 of the series is scheduled for Friday in San Antonio.
