San Antonio matched a season high with nine 3-pointers in winning its 18th consecutive home game against Atlanta and 10th straight overall against the Hawks.
Atlanta had season lows for points in any quarter (12 in the second), first-half points (37) and matched a season low for total points.
LaMarcus Aldridge added 13 points and Tim Duncan had 10 points and 18 rebounds for San Antonio, which has won 11 of 12.
Atlanta's starting five struggled to get a shot off against the hounding defense and long arms of Leonard, Aldridge, Duncan and Danny Green.
The Hawks' starters finished with 38 points. Reserves Mike Scott had 12 points and Kent Bazemore added 11.
San Antonio had six players score in double figures in remaining undefeated at home after nine games.
Aldridge was especially active early with eight points, five rebounds, two blocked shots and a steal in 10 minutes during the first quarter.
He jumped to corral a loose ball near the free throw line and missed a short jumper in the lane, but rose above three players to tip in his own miss.
The Hawks were 5 for 18 in the second quarter and committed six turnovers.
Hawks: Tiago Splitter, who sat out the game with a right hip injury, received a rousing ovation when he was shown on the midcourt video screens. Splitter played five seasons in San Antonio before being traded to Atlanta in the offseason in an effort to clear cap space to eventually sign Aldridge. ... Bazemore, who was listed as probable with a sprained right ankle, shot 5 for 10 in 20 minutes. ... Atlanta's previous low for points in any quarter was 18, which happened on three occasions.
Spurs: After Becky Hammon showed Gregg Popovich a play on the whiteboard she was carrying, he directed the second-year assistant to run the team's huddle during a timeout in the second quarter as he stood behind and watched. The team listened intently and the ensuing play resulted in an Atlanta turnover. ... Popovich was haranguing Marc Davis for nearly a minute before the official assessed a technical foul on the Spurs coach with 1:44 remaining in the first quarter. Popovich continued to lecture Davis following the technical, but Davis did not throw him out. ... San Antonio has not lost at home to Atlanta since drafting Tim Duncan with the No. 1 overall selection in 1997. The Hawks' previous win in San Antonio came on Feb. 15, 1997, when Steve Smith scored a game-high 25 points in a 109-89 victory.
Hawks: Host Oklahoma City on Monday.
Spurs: At Chicago on Monday.