Zimmermann (15-7) rebounded from an eight-run pounding in his previous start to control the Royals on a pair of runs, surrendering eight hits and a walk. The right-hander struck out seven.
Bryce Harper and Tyler Moore each contributed two hits with an RBI for Washington, which outlasted the Royals 11-10 in Friday's three-game series opener.
Kansas City's Wade Davis (6-10) was rocked for seven runs on eight hits over six innings.
Eric Hosmer and Mike Moustakas each drove in a run for the Royals, who dropped a seventh straight.
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