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.
Susan Sarandon 'very excited' about daughter's pregnancy
Dennis Rodman pledges to end trips to North Korea