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.
Celebrity Breakups and divorces of 2014 [PHOTOS]