Crawford hit for 27 points and Griffin carded 25 with 11 rebounds as the Clippers claimed a fifth win in six games overall.
DeAndre Jordan helped out with 10 points, 14 rebounds and three blocks for Los Angeles, which pulled away in the second half for the win.
Darren Collison totaled 11 points and six assists.
Enes Kanter led the Jazz with 23 points and 14 rebounds.
Richard Jefferson contributed 19 points.