The Rockets, who were down one point, 85-84, at the end of the third quarter, opened the fourth with a 15-1 run to take their biggest lead of the game, 99-86, with 7 minutes 32 seconds left.
The Clippers went on a 7-0 run to cut the deficit to four points, 109-105, with 37.8 seconds left after Jamal Crawford made a finger roll layup.
The final seconds of the game featured a series of missed shots and free throws, with DeAndre Jordan and Josh Smith each missing both of their attempts from the charity stripe.
The Rockets snapped a six-game losing streak to the Clippers with the victory.
The Rockets led over the Clippers at halftime, 62-60. Crawford (18 points) and J.J. Redick (12 points) combined for 30 points in the first half, but were held to just two points in the third quarter.
Crawford had a game-high 24 points. Chris Paul had 22 points and 14 assists and Jordan just fell short of his fourth 20-20 performance of the month, finishing with 22 points and 19 rebounds.
Harden led the Rockets with 21 points, most of which came from the free-throw line, from which he was 10 for 12. He shot only four for 13 from the field.
The Rockets didn't use the Hack-A-DJ strategy they used when the teams played Feb. 11. In that 110-95 Clippers win, Jordan shot 26 free throws, making 12 of them, finishing with 24 points and 20 rebounds.
In Wednesday's game, Jordan shot only eight free throws, of which he made four. The Rockets outshot the Clippers from the free-throw line, 30-19. The Rockets also forced the Clippers to commit 16 turnovers while only having 10 of their own.
Blake Griffin (elbow surgery to remove a staph infection) and Dwight Howard (right knee edema) missed Wednesday's game. The Clippers are now 4-3 without their superstar power forward.