Reddick's two-bagger sent home Yoenis Cespedes and John Jaso, opening the gates to a game-deciding, five-run outburst.
The A's right fielder struck again in the third with a two-run shot, handing Oakland a 7-1 lead.
Jed Lowrie went 3-for-5 with a bases-empty clout and two RBI while Eric Sogard also went deep en route to a three-RBI performance as the American League West-leading A's took two-of-three from Kansas City in the weekend series.
A.J. Griffin (7-6) held the Royals to two runs on five hits over five innings, earning the victory. He walked one and struck out four.
Kansas City's Luis Mendoza (2-5) lasted only 1 2/3 frames, surrendering five runs on five hits.
Alex Gordon and George Kottaras each homered for the Royals.