Williams’ basket tied the score at 103 with 1:58 to play and he gave the Clippers a 106-105 lead with a bucket with 50 seconds remaining.
Mills countered with a 3-pointer to give the Spurs a 108-106 lead with 37 seconds left, but Rivers answered by drilling a 3-pointer to give Los Angeles a 109-108 lead. Williams and Harris hit four free throws in the final seconds to secure the win.
The Clippers trailed by 19 points in the second quarter, but rallied and took a 64-63 with 5:29 remaining in the third quarter, following a three-point play by Jordan.
However, Aldridge scored nine points over the final 4:24 of the third quarter as the Spurs built an 83-72 lead after three periods.
The Clippers defeated San Antonio for the first time in three tries this season.
Williams is averaging 22.7 points and leads the NBA with 595 fourth-quarter points this season.
San Antonio fell to 14-25 on the road.
San Antonio led 31-17 after the first period. The Spurs increased their lead to 49-30 but the Clippers countered with a 19-7 run over the final five minutes of the quarter to cut the lead to 56-49 at halftime.
Harris made seven of 12 shots and led the Clippers with 17 points in the first half. Both teams shot over 45 percent in the opening half.
San Antonio: Since joining the NBA in 1976, the Spurs have accumulated the best winning percentage in NBA history (.623) . Entering Tuesday’s game, the Spurs led the league in fewest points scored by their opponents per game (99.2).
Clippers: Harris is averaging career-highs in points per game (18.6) and in 3-point percentage shooting (41.6). . The Clippers are 40-126 all-time against the Spurs. . F Danilo Gallinari missed Tuesday’s game with a hand injury.
Spurs: Remain in Los Angeles and play the Lakers on Wednesday.
Clippers: Travel to Salt Lake City to play the Jazz on Thursday.