The Royals' best player, Sweeney signed a five-year contract extension last week. On Friday, he hit his first homer of the season in the third inning, snapping a 2-2 tie.
Byrd (1-0) settled down after a shaky opening inning and allowed two runs and nine hits. The veteran righthander struck out two without a walk.
Blake Stein followed and gave up one hit over the final three innings for the save.
Raul Ibanez delivered a two-run double in the first for Kansas City, which began the season with back-to-back home losses to Minnesota.
Chicago starter Jon Garland (0-1) surrendered five runs -- four earned -- and six hits over five innings. He walked four without a strikeout.