Blair scored 15 of his points in the final quarter, including a nifty basket inside with 1:02 left as the Spurs erased an 11-point deficit over the final 12 minutes.
George Hill added 12 points and Michael Finley scored 10 for San Antonio (1-1 preseason), which played without Tim Duncan and Manu Ginobili.
Daequan Cook scored 20 points, Mario Chalmers added 17 and Michael Beasley finished with 16 for Miami (0-3). The Heat were without several regulars, including reigning NBA scoring champion Dwyane Wade.
Cook tried to win it with a 3-pointer at buzzer, but his shot bounced off the rim.
Blair's steal and layup with three minutes left helped the Spurs take control, and further illustrated why he's looked like the steal of the NBA Draft so far in preseason. The 37th overall pick last June had 16 points and 19 rebounds in San Antonio's only other preseason game so far, not including the Spurs' win in an exhibition last week over Greek club Olympiakos.
Both sides were far from being full strength, with at least two probable starters unavailable for each team.
For the Heat, Wade (left ribs), Jermaine O'Neal (left foot) and Quentin Richardson (left ankle) were among those sitting out. The Spurs opted to leave Duncan and Ginobili at home to rest.
Wade went through a pregame workout on the Heat practice court, then received treatment. Heat coach Erik Spoelstra said the team doesn't believe the muscle strain will be a long-term concern for Wade.
"He's feeling better," Spoelstra said. "He was actually able to do quite a few exercises in the weight room [Saturday] and that's going to be part of the proactive approach that we take with him to try to gain more flexibility in this area and try to build up some strength."
Miami was also without backup point guard Chris Quinn (right foot). The Heat - who insist that they're committed to Chalmers being the starter again - held internal discussions Saturday about adding depth at that position, and Spoelstra said more talks will take place Monday.
Chalmers scored 11 of his points in the third quarter, when Miami stretched its lead to 73-62 entering the final 12 minutes.