Jimenez (3-2) allowed only three hits while walking one and striking out eight en route to his third win in three starts for the Indians, who evened their three-game series.
Asdrubal Cabrera went 3-for-5 with two RBI, including a sixth-inning RBI single that put Cleveland ahead 5-1.
Mike Aviles added another run-scoring hit in the seventh for a 6-1 advantage and the Indians would need all those runs as the Tigers came back for four in the bottom of the seventh, highlighted by Omar Infante's two-run triple off reliever Nick Hagadone.
Jason Giambi's sacrifice fly scored Jason Kipnes in the eighth, handing the Indians a 7-5 advantage.
Tigers starter Justin Verlander (4-3) was tagged with the loss after allowing four runs (three earned) on six hits in five frames.
Jhonny Peralta homered for Detroit.