Leonard finished with 24 points and six rebounds, while Paul George had 19 points and eight rebounds for the Clippers, who won their fourth consecutive game on Friday, AFP reported.
"It feels great. We are trying to build something," Leonard said.
"We made a few little mistakes in the first half, but we ended up closing it out in the fourth though.
"We are building. This is just our second game with everyone playing again and it has just been great."
Clippers' Lou Williams scored all 26 of his points in the second half in front of a crowd of 19,000 at STAPLES Center arena.
James Harden had 37 points, 12 assists and eight rebounds for the Rockets, who have lost their last two games after going on an eight-game winning streak.
Williams gave the Clippers a 118-117 lead with 30 seconds remaining by draining a three-point shot. Harden then made two free throws to restore Houston's lead, 119-118, with 22 seconds to go.
After Leonard's bucket, George helped clinch it by making two free throws with one second on the clock.
It was the first meeting between Houston's Russell Westbrook and George. They played two seasons together with the Oklahoma City Thunder before George left to join the Clippers.
The Thunder dealt Westbrook to the Rockets a short time later.
Westbrook scored 22 points but he missed a three-point attempt late in the game which would have given the Rockets a shot at victory.
Elsewhere, LeBron James had 23 points and Anthony Davis made four clutch free throws in the final ten seconds as the Los Angeles Lakers pulled out a 130-127 win over the Oklahoma City on Friday.
Davis finished with a team-high 33 points and 11 rebounds, but it was his free throw shooting that helped the Lakers seal the victory over the struggling Thunder.
Chris Paul made a layup with 11 seconds remaining to get the Thunder to within 126-125 before Davis made a pair of free throws to put the Lakers up by three with eight seconds left.
Shai Gilgeous-Alexander scored with four seconds to go before two more free throws from Davis gave the Lakers another three-point cushion. The Thunder made a final long desperation inbounds pass, but James stole the ball from Steven Adams to end the game.
"We find a way to win no matter what the game is," said James.
"This is a very good team at home. They've struggled on the road a little bit, but we knew they were going to give it their best punch."
This marked the Lakers' second win over the Thunder in a span of three days, after beating Oklahoma City 112-107 on Tuesday at Los Angeles. The Thunder have now lost three straight and five of their last six.
The Lakers won their sixth straight game and earned a 13th victory in their past 14 contests.
It is the seventh consecutive game that James has posted double-digit assists. He leads the league with an average of 11.3 assists per game.
Gilgeous-Alexander scored 24 points and Adams added 22 for the Thunder.
Paul added 18 points and seven assists for Oklahoma City, which tied a season-high in points.
Also, Luka Doncic posted his third straight 30-plus point game as the Dallas Mavericks swept all four games on their recent homestand with a 143-101 rout of the Cleveland Cavaliers.
Doncic, who rested on the bench for the entire fourth quarter, finished with 30 points and 14 assists on 11 of 18 shooting from the floor.
He is the eighth Mavericks player to score 30-plus points in three straight games, joining among others Dirk Nowitzki who did it in 2010.
Justin Jackson had 19 points, Kristaps Porzingis scored 17, and Tim Hardaway had 16 for Dallas, who had seven players finish in double-digit scoring.
Darius Garland scored 23 points for Cleveland, who have lost six in a row.