"It was a great win for us against the really active team," Spurs coach Gregg Popovich said.
LaMarcus Aldridge clinched the Spurs come-from-behind victory with an offensive rebound and two free throws with 17 seconds remaining.
The Spurs, who trailed by 14 points in the first half, improved to 13-3 on the season in front of a crowd of 18,600 Friday at the Boston Garden arena.
San Antonio rallied for their eighth straight NBA win, their ninth in as many games on the road this season, and also beat Boston for the 10th straight time.
Leonard, who finished with 25 points, 10 rebounds and four assists, scored nine points in the fourth quarter.
Former Celtic Patty Mills tallied a season-best 19 points and rookie Davis Bertans and veteran David Lee added 15 points apiece.
That marked a career high for Bertans and season high for Lee, who also grabbed 12 rebounds.
Starters Danny Green and Aldridge scored 11 and 10 points, respectively.
Isaiah Thomas finished with 24 points and eight assists, while Avery Bradley had 19 points and eight rebounds for the Celtics who had their three-game winning streak snapped.
Al Horford added 12 points, 10 rebounds and three blocks while Marcus Smart came off the bench to score nine points.
"I felt like we fought really hard all night," said Horford. "I think the stat sheets says their bench was the difference. They did a great job."
The Celtics used an 18-2 first-quarter run to grab a 24-10 lead.
Boston, coming off a three-game road trip, then went 5:51 without a point in the second quarter as San Antonio used a 12-0 burst to get back in the game.
Meanwhile, Cleveland and Kevin Love remained hot from the three-point line as the Cavaliers routed the Dallas Mavericks 128-90 in record fashion.
Love finished with 27 points and 10 rebounds, Kyrie Irving scored 25 points and LeBron James had 19 points and 11 assists. None of them played the fourth quarter in a one-sided contest that was decided by halftime.
The Cavaliers also set an NBA record by becoming the first team to have consecutive games with 20 or more three pointers. Cleveland made 21 three pointers on Wednesday in a lopsided win over the Portland Trail Blazers.
- Love hot from beyond the arc -
Love drained seven three pointers Friday, including one that gave Cleveland a whopping 68-28 lead in the second quarter.
One game after he scored 34 points in the first quarter against Portland, Love added to his three-point total Friday, giving him 15 in the last two games. He made 18 through the first 12 games of the season.
In New York, Carmelo Anthony's jumper with three seconds left in overtime gave the New York Knicks a 113-111 win over the Charlotte Hornets at Madison Square Garden.
Anthony poured in a season-high 35 points and grabbed 12 rebounds and Kristaps Porzingis and Derrick Rose scored 16 points each for the Knicks.