With the game tied 2-2 and one out in the seventh, Rios crushed a 3-0 pitch from Royals starter Luke Hochevar (8-14) down the left-field line for a long homer.
The Tigers bested Oakland 12-2 earlier Tuesday.
Gavin Floyd (10-10) earned the win with seven innings of two-run ball, yielding five hits while walking one and striking out three.
Matt Thornton tossed 1 1/3 scoreless relief innings and Addison Reed recorded the final two outs for his 28th save.
Hochevar ended up allowing three runs on seven hits over seven full frames for the Royals, who dropped the opener of a three-game series.
Billy Butler drove in both Kansas City runs.