The win evened the series, which will be decided in Game 7 Thursday in Miami.
James completed his second triple-double of the year with 11 assists and 10 rebounds to go with his 32 points. James hit a 7-foot jumper with 1:43 left in OT to put Miami ahead 101-100. His heroics also included a late long-range bucket in regulation, which set the stage for a 3-point shot by Ray Allen that forced overtime.
San Antonio had led by five with 28.2 to play in regulation but could not hold off the Heat. Miami outscored the Spurs 30-20 in the fourth quarter and 8-5 in OT.
Tim Duncan led San Antonio with 30 points, although 25 of them came in the first half. Kawhi Leonard added 22.
The Spurs shot 43.5 percent from the field and were 5-of-18 from 3-point range. Miami shot 46.9 percent from the field and hit 11-of-19 from outside the arc.