Thome appeared in a Cleveland uniform for the first time since 2002, having waived a no-trade clause in his Minnesota contract so the Twins could deal him to the Indians. He appeared as the Cleveland designated hitter and went hitless in four at-bats, striking out twice.
Jimenez (2-1) gave up a run on seven hits, the lone Kansas City run coming on a fourth-inning homer by Eric Hosmer.
Cleveland rallied for two runs in the seventh, getting an RBI single from Jack Hannahan to tie it. Ezequiel Carrera drew a bases-loaded walk later in the inning to give the Indians their second win in eight games.
Chris Perez allowed the potential tying run to reach third in the ninth before picking up his 28th save.
Felipe Paulino (2-6) took the loss, giving up the two runs on six hits in 6 2/3 innings.