San Antonio used a 16-0 run bridging the end of the third quarter and the first two-and-a-half minutes of the fourth to take an 83-74 lead.
The Suns fought back and went in front on a pair of free throws by Devin Booker with 52 seconds to play.
But Sochan, who will be included in Poland's national team for the upcoming Olympic qualifiers, poured in his deciding shot from beyond the arc with 29.2 seconds left.
Kevin Durant and Booker of Suns both missed 3-pointers in the final sequence,
allowing the Spurs to escape with the unlikely win.
Sochan also grabbed a career-high 18 rebounds for San Antonio. Devin Vassell added 26 points while Zach Collins had 18 and Keldon Johnson scored 14 for the Spurs (16-56), who snapped a three-game losing streak but are still in last place in the Western Conference.
The Spurs had to do without star rookie
Victor Wembanyama, who sprained his ankle just before game time.
