The 24-year-old Portugal international was signed by City for about £65m from Benfica last summer.
He has helped the club win the Premier League and Carabao Cup as well as reach the final of the Champions League, in which they will play Chelsea.
“It is a huge privilege. Obviously I could not have done it without the success of the team,” said Dias.
“I think me receiving this prize is the major example of how our team works, the way we build our game.
“It reflects all the togetherness we have on the pitch, how we perform and the spirit that is in the team and with these players.
Dias is only the third player to win the FWA award in his first season in English football, and the last defender to win the accolade was Liverpool’s Steve Nicol 32 years ago.
Chelsea and England forward Fran Kirby won the FWA women’s award earlier this month.
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