The game was a makeup from an April 14 rainout.
Gee (9-8) allowed just four hits through five innings and was lifted with two outs in the eighth after recording his ninth strikeout of the game.
The only run he was charged with came in the seventh when he struck out the first two batters and Trevor Plouffe doubled. Mets second baseman Daniel Murphy then misplayed a grounder for an error on which Plouffe scored.
The Mets were already up 5-0 at that point, however, getting to Twins starter Kyle Gibson (2-4) for four runs and 10 hits in 3 2/3 innings.
Andre Brown delivered a two-out RBI single in the first for New York and Eric Young had the third of three consecutive singles for the Mets to drive in a run in the second. Omar Quintanilla and Murphy had run-scoring singles in the fourth for a 4-0 lead that chased Gibson from the game.
Wilmer Flores made the score 5-0 in the seventh with another RBI single and Marlon Byrd completed the scoring with a home run in the ninth. It was his 20th homer of the season.
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