The Spurs had won five in a row and 16-of-17 before running out of steam on the next-to-last stop of a nine-game road trip. The journey will end on Sunday in Phoenix.
San Antonio built a 13-point lead in the fourth quarter, but after a Golden State rally it took a layup by the Spurs' Manu Ginobili with 4.5 seconds remaining to force overtime.
The Spurs moved in front at 98-96 in the extra period before the Warriors went on their late rush.
They scored six quick points to take the lead with Jack hitting back-to-back baskets and he closed out the surge by making three free throws over the final 12 seconds.
Jack finished with 30 points while David Lee had 25 points and a massive 22 rebounds for the Warriors.
Danny Green led the Spurs with 20 points and was one of six San Antonio players in double figures.