The win snapped Golden State's 16-game losing streak to San Antonio. The Warriors last beat the Spurs in 2008.
Jack came up big in the fourth quarter and overtime, scoring 17 of his 30 points. He finished the game with 10 assists.
David Lee broke loose for 25 points and 22 rebounds for the Warriors, who trailed by 13 points midway the fourth quarter.
Stephen Curry added 19 points.
The come-from-behind victory ended a five-game winning streak by the league-leading Spurs, who have not played a home game since February 2.
Tim Duncan had 19 points and 13 rebounds for San Antonio with Danny Green finishing with 20 and Tony Parker 18.
It took a layup by San Antonio's Manu Ginobili on a pass from Duncan with four seconds left in regulation to force overtime.
Then, after Tiago Splitter's inside shot had given San Antonio its last lead with two minutes remaining in the extra session, Golden State scored in streaks.
The Warriors ran off six consecutive points, four by Jack, and never trailed again, scoring the game's final five points on free throws after San Antonio had pulled with one.