After trailing by as many as 10 points, San Antonio forced seven turnovers while outscoring Phoenix 30-16 in the final quarter to capture its sixth straight win at home.
Tony Parker added 20 points and Pau Gasol had 16 points and 10 rebounds for San Antonio.
The Spurs were without Leonard for the first time this season. The 6-foot-7 forward, who has a stomach virus, was replaced in the lineup by Kyle Anderson.
T.J. Warren led Phoenix with 23 points, going 10 for 10 on free throws. Brandon Knight added 14 points and Marquese Chriss and Eric Bledsoe each had 13.