Kershaw put himself in trouble early, allowing a runner in scoring position with one out in the first inning and with nobody out in the second. But after Rickie Weeks led off the second frame with a Matt Kemp-aided two-bagger, Kershaw retired the next 18 straight batters.
Crawford homered to lead off the first inning, then added another solo shot in the fifth inning. It was the sixth two-homer game of Crawford's career, and his first since July 8, 2010 while with the Rays, against the Indians.
Crawford joins Mark Ellis as the only two Dodgers with multi-homer games this season, and at four home runs Crawford took over the team lead.
Kershaw struck out 12 on the day, his 19th career game of 10 or more strikeouts and first this season. After walking eight batters in his last two starts combined, Kershaw did not issue a free pass on Sunday, the fifth game of his career with at least 10 strikeouts and no walks.
Milwaukee threatened in the eighth inning when Carlos Gomez opened with a double, but he was later erased on a heads-up fielders choice by Kershaw, a play that ended with Juan Uribe running down the speedy Gomez for the tag. Kershaw, fittingly, accounted for the other two outs in the inning with strikeouts.
Since 1916, there have been 18 games by a Dodgers starting pitcher with at least 12 strikeouts and no walks. The last two have been accomplished by Kershaw, who joins Sandy Koufax and Don Drysdale as the only repeat offenders on the list.
On the season, Kershaw now has 47 strikeouts, one behind A.J. Burnett for the major league lead, and 12 walks in 41⅔ innings, to go along with his 1.73 ERA.
Home runs: Carl Crawford 2 (4)
WP - Clayton Kershaw (3-2): 8 IP, 4 hits, 12 strikeouts
LP - Kyle Lohse (1-2): 7 IP, 5 hits, 2 runs, 4 strikeouts
Sv - Brandon League (8): 3 up, 3 down