The 22-year-old made 12 league starts for Newport after arriving in January and helped them avoid relegation.
He made 35 appearances in all competitions in the 2016-17 season, for a combination of his former side Mansfield Town and the Welsh outfit.
Rose, who will join Grimsby on 1 July, is the younger brother of Tottenham Hotspur and England left-back Danny.
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Grimsby Town have signed midfielder Mitch Rose from fellow League Two club Newport County on a two-year contract.