The Angels put one on the board early in the top of the second inning when Adam Kennedy singled to score Rivera.
Royals' Doug Mientkiewicz scored the only run for Kansas City in the bottom of the sixth inning.
Los Angeles' Jared Weaver gets the win and becomes the first rookie in 25 years to win his first seven starts.
The Royals have Monday off before hosting the Baltimore Orioles on Tuesday for a three-game series.
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