Jimenez (6-4) rebounded from a poor start against the Chicago White Sox in his previous outing to tame the Tigers on five hits, issuing one walk and striking out four.
Vinny Pestano tossed a scoreless eighth and Chris Perez nailed down his 18th save despite allowing two hits and a run in the ninth for the Indians, who took the opener of a three-game series.
Detroit starter Drew Smyly (2-2) absorbed the loss, yielding four runs on six hits over six innings.
Miguel Cabrera went 2-for-4 with a run-scoring double for the Tigers.
Celebrity Families of 2014 [PHOTOS]