Harden missed his third game of the season with a right ankle sprain, but the Rockets had all the right answers in his absence.
Harden, who played 26 minutes and scored a four-month low seven points in a 103-94 road loss to Memphis on Friday night, ranks fifth in the league at 26.0 points per game.
"We had a lot of guys play well," Rockets coach Kevin McHale said. "The ball was really moving tonight, and it was good to see that. Everybody was in tune."
Garcia had his best game since being traded to Houston from Sacramento in February, hitting 3 of 7 shots from beyond the arc as a spark off the bench to help Houston snap a two-game losing streak.
"When your best player is out, everybody has got to step up," Garcia said. "That's what we did tonight. We stayed together and got the win."
Lin and Parsons combined for 13 points in the third quarter, when the Rockets outscored Los Angeles 30-13. The rally gave Houston a 22-point lead heading into the fourth quarter, and Clippers coach Vinny Del Negro had seen enough.
"We were terrible, our effort was terrible, our attitude was terrible and our urgency was terrible," Del Negro said. "They know exactly how I feel, and they didn't say much. That's not what we're about."
Los Angeles was playing its fourth game in five nights, but Del Negro said the schedule was no excuse for Saturday night's display.
For the second consecutive game, the Clippers failed to secure their franchise-record 50th victory. The Clippers tied the 1974-75 team record for wins in a season by beating New Orleans on Wednesday.
Paul led the Clippers with 19 points, seven assists and three steals.
"That was terrible," Paul said. "If I knew why, I'd tell you. But I know this is on us."
Los Angeles has lost three of its last four games, allowing at least 98 points in each loss.
The Rockets frustrated the Clippers in the second quarter with Griffin and Jamal Crawford both drawing technical fouls for disputing calls against them.
"When that happens, it's a selfish act that doesn't make any sense, and it hurts the team," Del Negro said of the technicals. "I've got a lot of whining going on and a lack of urgency going on and it's disappointing."
The Rockets outscored the Clippers 48-26 in the paint, and the Houston bench outscored Los Angeles' reserves 39-31. Carlos Delfino scored 11 points and Patrick Beverley added nine for the Rockets' bench.
Garcia hit three straight 3-pointers in the first half to lead the Rockets with 13 first-half points. Houston led 55-50 at the break.
"If you look across the board, no one really did anything spectacular," Lin said. "But everybody played their role, did their jobs, and that's what you need when you're playing without your star player."
Asik registered his 30th double-double of the season with 13 points and 12 rebounds, adding two blocks.
Making his first NBA start in place of Harden, Rockets guard James Anderson scored seven points and added five rebounds, two steals and two assists. Rockets forward Greg Smith had nine points and eight rebounds, also in his first NBA start.
The loss dropped Los Angeles to fifth in the Western Conference as Memphis moved into fourth with a win over Minnesota.
Houston finished March with a 9-4 record after going 6-5 in February. On Monday, the Rockets host Orlando, while the Clippers host Indiana.
Harden sprained his right ankle in a Feb. 12 road victory at Golden State and did not play the following game. A similar injury to the same ankle also caused him to miss a Dec. 10 meeting against San Antonio. ... Clippers forward Matt Barnes left the game in the fourth quarter with a left ankle sprain. ... Parsons, a University of Florida graduate, said he plans to attend Sunday's South Regional final between the Gators and Michigan, with a trip to the Final Four on the line.