Reddick laced a pitch into the right center-field gap to make to it 4-1 Oakland in the fifth and then came home on a Yoenis Cespedes single to key the decisive frame.
Kila Ka'aihue's single added another run and Adam Rosales contributed a two-run hit in the ninth as Oakland logged its first win since May 21.
Brandon McCarthy (4-3) worked around eight hits over six innings to earn the victory, allowing a pair of runs while walking two and striking out two.
Royals starter Luke Hochevar (3-6) was tagged for all six runs on five hits in a 4 2/3-inning effort.
Jeff Francoeur went 2-for-4 with an RBI and Jarrod Dyson added three hits for Kansas City.