Kershaw (5-2) turned his sixth consecutive start without recording a loss, stifling the Brewers on one run while issuing a single walk and striking out five in the 107-pitch gem.
Brewers starter Yovani Gallardo (3-4) absorbed the loss, surrendering three runs on eight hits over six full frames.
Norichika Aoki got the only RBI for Milwaukee, which fell to 3-15 in May.
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