The Twins were trailing 3-0 in the eighth when they shaved the deficit to one run on RBI doubles from Eduardo Escobar and Ryan Doumit.
Dozier then tied it with a lead-off solo homer in the ninth off Joaquin Benoit.
The game remain knotted until the 11th when Dozier collected a lead-off single against Luke Putkonen (1-2), went to second on a wild pitch and scored the game-winner on Pinto's solid single to right.
Anthony Swarzak (3-2) earned the win for Minnesota, which ended a four-game losing streak.
Torii Hunter went 3-for-6 with an RBI for American League Central-leading Detroit, whose magic number to win the division crown remained at two.