The Astros took a 3-2 lead in the top of the ninth on an RBI double by Matt Dominguez, but Dozier delivered Doug Bernier with a base hit in the bottom half of the inning to extend the game.
Minnesota ended a four-game losing streak, using seven pitchers in the process.
Ryan Pressly (3-2) worked the 12th and 13th innings to get the victory, allowing two hits and striking out one.
The winning run came off Dallas Keuchel (5-6), who was the eighth Houston pitcher to see action.
Celebrity Families of 2014 [PHOTOS]