Danks (3-10) hadn't won a game since July 2, going eight starts without a victory until handling the Royals on seven hits over eight scoreless innings Tuesday. The left-hander walked one and struck out two.
Addison Reed finished up with his 32nd save as Chisox took the first of a three-game series against their division rivals.
Kansas City's Ervin Santana (8-7) absorbed the loss despite yielding just one earned run on four hits over six innings.
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