Trailing 3-2 in the ninth, the Tigers battled back against Twins closer Matt Capps (0-3).
Quintin Berry, who made a spectacular, diving catch in center field to end the eighth inning, led off the ninth with a walk, stole second and scored when Cabrera unloaded on an 0-2 pitch, sending it over the fence in straightaway center field.
Berry ended the game 3-for-4 with two runs scored.
The rally handed the victory to Brayan Villarreal (1-1), who recorded the final two outs of the eighth, and Jose Valverde nailed down his ninth save as the Tigers completed a three-game sweep of the Twins.
Alexi Casilla went 3-for-3 with an RBI for Minnesota, which dropped its fifth straight overall.