San Antonio won its eighth straight game and matched Indiana's 10-1 record as the best in the NBA.
Boston lost its first four games of the season, then won four in a row and now has lost five straight.
The Celtics owned a two-point lead early in the second half when the Spurs went on a 13-2 outburst. Leonard had 12 of his 16 points in the third period and combined with Tim Duncan to score all of the points during the run.
Tony Parker had 19 points to lead the San Antonio scorers while Duncan had 13 points and nine rebounds.