Jimenez (11-9) picked up his third win in three starts, yielding just one unearned run. The right-hander didn't issue a walk and struck out 10 in the 99-pitch performance.
He exited with a 3-1 lead and the Indians tacked on another run in the bottom of the seventh on a Carlos Santana homer, but then saw their lead dwindle when Kansas City's Alex Gordon pounded a two-run shot in the eighth off Cody Allen.
Closer Chris Perez issued a pair of walks and a hit in the ninth to load the bases with two outs, but then retired Gordon on a fly ball to end the contest and pick up his 23rd save.
Asdrubal Cabrera and Yan Gomes also homered for the Tribe, who moved to within 1 1/2 games of idle Tampa Bay for the second American League wild card spot with a victory in the opener of a three-game series.
Ervin Santana (8-9) surrendered four runs (three earned) on four hits over seven innings for the Royals, who dropped four games behind the Rays in the wild card chase.