Kazmir (4-4) gave up a run on five hits over seven innings, striking out seven.
The Indians gave Kazmir a 2-1 lead heading into the bottom of the seventh, then produced some insurance in that inning when Drew Stubbs singled in a run and Kipnis delivered two more with a base hit.
Kipnis also had a sacrifice fly in the third to give Cleveland a 2-0 advantage.
Samuel Deduno (3-2) took the loss, giving up four runs on six hits in six innings.
The Twins had been trying for their fourth straight victory, which would have equaled their longest winning streak of the season.
Celebrity Families of 2014 [PHOTOS]