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Hicks sent the full-count fastball from Red Sox reliever Andrew Miller to left field, while Suzuki sped around the bases to score the game-winning run.
The Twins (19-20) have taken two of three games from the defending World Series Champion Red Sox.
Brian Deunsing, who pitched the top of the 10th inning, picked up the win. Miller took the loss.
Four singles in the ninth inning off of Twins closer Glenn Perkins allowed Boston to erase a 3-1 deficit and tie the game in the top of the ninth. It was the second blown save of the season for Perkins.
The Twins got to Red Sox starter Clay Buchholz in the second inning. Chris Parmelee hit his second home run of the series. He also won Tuesday night's game with a walk-off home run.