The Indians seemed on their way to breaking their long skid when they took a 5-1 lead into the seventh, but the Twins scored three runs -- including two on an error by second baseman Jason Kipnis -- to pull within one.
That brought up Justin Morneau, whose grounder to first base was booted by Casey Kotchmann, allowing Mastroianni to score the tying run.
Morneau went to third on Ryan Doumit's double and Nishioka followed with a deep fly to left, chasing him home with the go-ahead run.
Brian Dozier completed the rally with a run-scoring single and the Twins went on to clinch their three-game series.
Tyler Robertson (1-0) earned his first major league victory with two scoreless frames and Glen Perkins collected his seventh save for Minnesota, which improved to 5-2 in August.
Shelley Duncan hit a two-run homer for Cleveland, which dropped its seventh straight to the Twins.