On March 7, 2015, Young signed with the Kansas City Royals. Young entered the season in the long reliever role with the Royals. He appeared in the Royals' sixth game on April 12 against the Los Angeles Angels with two scoreless innings as the Royals started the season 6–0. Young made his first start with the Royals on May 1. He tossed five no-hit innings against the Detroit Tigers. On June 16, Young posted 7 shutout innings and 3 runs batted in against the Milwaukee Brewers, becoming the first Royals pitcher to tally 3 RBI in a game since 1972. After achieving a record of 8–6 with a 3.25 ERA through the end of July, Young was returned to the bullpen on July 31 after posting a 5.11 ERA over a 5-game stretch. On September 27 (one day after his father died), Young made his first start since July 28 and pitched 5 no-hit innings against the Cleveland Indians. On October 20, Young started and pitched 4 2/3 innings in a game 4 victory in the 2015 American League Championship Series against the Toronto Blue Jays. It was Young's first postseason start since the 2006 National League Division Series. He earned the win in the October 27 game 1 of the 2015 World Series when he shut down the New York Mets, giving up a walk and no hits while striking out 4 over the final three innings, to help the Royals win 5–4 in 14 innings.