Kershaw (3-2) allowed four hits and no walks in a 117-pitch outing, retiring 16 straight batters between the second and seventh innings while striking out 12 overall.
Brandon League earned his eighth save with a 1-2-3 ninth.
Crawford homered in the first and fifth innings.
Kyle Lohse (1-2) took the hard-luck loss for the Brewers, allowing just the two homers among five hits over seven innings. He struck out four.