The Mavericks trailed by one heading into the fourth period and then surged to an apparent victory when they built a 17-point lead with 4:49 to go.
Los Angeles, however, outscored Dallas the rest of the way 23-4 and held the Mavericks scoreless for the final 2:57.
Dallas missed its last six shots, four of the attempts coming from Dirk Nowitzki, and the Mavericks turned it over on their final possession.
Crawford's two free throws with 11 seconds remaining gave the Clippers a 128-127 lead, after which Jose Calderon missed a 3-point try.
The Clippers won their fourth in a row and ended the Dallas winning streak at three.
Nowitzki scored 27 points and grabbed eight rebounds, but he made only 8-of-22 shots from the field.