Jimenez (13-9) allowed five hits and a run for the Indians, who nailed down the first American League wild card spot and will host a Wednesday home game against either Tampa Bay or Texas.
The Rays and Rangers will face off against each other Monday in a wild card tiebreaker.
Nick Swisher slugged a two-run homer in the first inning as Cleveland (92-70) swept the four-game series from the Twins -- their seventh such sweep this season.
Scott Diamond (6-13) was reached for four runs (two earned) in six innings for the Twins (66-96), who ended the season with six straight losses.
Celebrity Families of 2014 [PHOTOS]