Hosts Phoenix led by 13 points early in the fourth quarter, but Victor Wembanyama’s jump shot reduced their lead to 114-111 with 49 seconds to play.
The number one draft pick dunked from a rebound with six seconds left, before Keldon Johnson stole the ball from Kevin Durant to score and give the Spurs their only lead of the game with 1.2 seconds remaining.
A minute earlier, Durant became the 12th player in NBA history to reach 27,000 points.
The 35-year-old, who scored 26 points at Footprint Center, reached the landmark in his 990th game. Only Wilt Chamberlain, Michael Jordan and Kareem Abdul-Jabbar have done so in fewer matches.
Johnson scored 27 points, while French rookie Wembanyama, playing only his fourth NBA game, and Devin Vassell each added 18 as the Spurs recovered from Sunday’s 40-point loss at the Los Angeles Clippers to claim their first away win this season.
The Spurs are eighth in the Western Conference and the Suns 10th. They meet again at the same venue on Thursday.
George missed all six of his first-quarter shots, but made four three-pointers in scoring 14 points in the third quarter.