The Royals clubbed 16 hits, but they could not keep up with the Indians in the power department and moved within two games of falling into the American League Central cellar.
Cleveland scored six times in the second inning. After LaPorta's three-run blast, Gimenez had an RBI double and both Michael Brantley and Chin-Soo Choo had run-scoring singles.
Gimenez hit his homer in the fifth, his first of the year and fourth of his major league career. Nix tacked on his three-run shot in the eighth.
Josh Tomlin (2-3) lasted five innings to get the win, allowing three runs on eight hits with four walks.
Bryan Bullington (1-3) was roughed up for seven runs on nine hits in four innings.
Cleveland had scored just seven runs in its previous six games.