Wade had scored no more than 21 points in any of his previous 18 post-season contests this season, but he hit 14-of-25 from the field and delivered 32 points in Game 4 of the championship series.
Miami regained home-court advantage by pulling even at 2-2. San Antonio will host the pivotal fifth game of the series on Sunday before the teams return to Florida for the sixth game on Tuesday. A seventh game, if needed, will be hosted by the Heat next Thursday.
LeBron James, answering criticism for his lackluster play in this series, scored 33 points with nine of those coming in the closing minutes after the game had been decided.
It was Wade, however, who carried the Heat as they pulled away in the early stages of the fourth quarter.
San Antonio was within five to open the final period, but Miami began the quarter on a 13-7 run during which Wade scored eight points.
Miami shot 53 percent from the field and forced 18 San Antonio turnovers.
After hitting an NBA Finals record 16 3-pointers in Game 3, the Spurs were held to eight on Thursday. Five San Antonio players scored in double figures, with Tim Duncan producing 20, but the Spurs could not slow down the Miami offense.