Los Angeles outscored the Heat 13-4 over the final 3:45 of the third quarter, took an 11-point lead into the final 12 minutes and increased the advantage to 20 inside the 6-minute mark.
Miami fell to 2-2 on its six-game road trip, which will conclude with stops in Denver and Phoenix.
Crawford came off the bench to make 7-of-11 from the field and all six of his free-throw attempts for 22 points. Blake Griffin had 20 points and 14 rebounds, Chris Paul added 16 and Caron Butler had 15.
The Clippers shot 48 percent and made 10-of-21 from 3-point range.