In the closing seconds, Clippers center DeAndre Jordan stuffed Curtis Jerrells' layup attempt that would have given the Spurs the lead, then hit two free throws with 5.1 seconds left. Jerrells missed a 3-pointer at the buzzer.
Griffin scored 10 points in the first quarter and had 13 in the third.
"It was cool," Griffin said of his first-quarter, head-to-head matchup with perennial All-NBA post Tim Duncan. "I've watched him for as long as I can remember, since I was 8 years old and he was in his first year in the league. It was really cool to go up against him."
The Clippers outscored the Spurs 29-26 in the fourth quarter behind 13 points from Al Thornton.
Richard Jefferson, acquired by San Antonio from Milwaukee in an offseason trade, scored 15 points in 16 minutes, all in the first half. Rookie DeJuan Blair added 11 points and 12 rebounds for the Spurs.
"I came into a situation where a lot of guys already have known the system for a long time," Jefferson said. "My job is to learn the system and figure out how it all fits for me. Once I can figure that out, I can be more aggressive. It's almost like I am a rookie again, but the more I get to play, the more aggressive I can be."
Jordan, a second-year player from nearby Texas A&M, finished with 15 points and eight rebounds.
Neither Jefferson nor Tim Duncan, who had six points and five rebounds in 17 minutes, played in the second half. Tony Parker did not play at all for the Spurs.
Jefferson scored eight straight points for the Spurs during a late stretch in the second quarter. One of his baskets in the run came off crisp passes from Manu Ginobili and Antonio McDyess, another heralded offseason pickup as a free agent.
The Spurs took their first lead, 28-27, when Jerrells hit a 21-foot jumper not quite 3 minutes into the second quarter. Jefferson re-entered the game at the 8:20 mark and took over, scoring the eight straight points in a 2-minute stretch.
George Hill's 3-pointer with 29.4 seconds left in the first half gave the Spurs a 51-44 lead. Hill finished with 13 points.
Griffin, who didn't play in the second quarter, had a thunderous dunk at the beginning of the third. Later in the quarter he converted another alley-oop, this one one-handed off a pass from Ricky Davis.
"We have to make plays like that," said Griffin, who was 5 of 6 from the field in the third period and 10 of 14 overall. "It was great that we kept it somewhat close throughout the game, and we made the plays that gave us the momentum."
The lead went back and forth in the final quarter before Thornton scored 11 straight points for the Clippers, including a 3 that put them up 89-83.