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Dozier, who earlier in the day was snubbed for the All-Star Game, highlighted the victory. It was the lone hit for Dozier of the night, who is one of five finalists in a fan vote for the final roster spot on the American League team.
The home run was Dozier's 17th of the season. He has also driven in 42 runs and scored an American League leading 63 runs on the season.
Baltimore took the early lead on the game's second pitch when Manny Machado homered to center field. But it didn't last long, as Torii Hunter tied it up in the bottom of the firstwith a solo shot of his own.
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The Twins made it 2-1 in the second inning, when Aaron Hicks went deep for the second time in as many days. But Baltimore returned the favor with a solo homer from Adam Jones in the sixth to even it up again.
Twins starter Phil Hughes battled through 6 1/3 innings. He gave up two runs on nine hits, while striking out three. He didn't factor in the decision. The win went to Trevor May (5-7), who pitched a scoreless inning of relief in the top of the 10th.
Kyle Gibson takes the mound for the Twins on Tuesday night against 24-year-old righty Kevin Gausman.