Nicolas Otamendi has also completed a move in the opposite direction to Benfica in a deal worth £14m ($17m).
Dias, 23, who could cost a further £3.3m with add-ons, says he cannot wait to work with City boss Pep Guardiola, a man he believes can improve his all-round game.
“To have the opportunity to join a club like Manchester City is fantastic for me and one I couldn’t turn down,” he said.
“Their success speaks for itself. They have been the dominant team in England over the last few years, playing an attacking brand of football which I feel suits my own game.
“It is really exciting to be part of such a talented squad and to play for a world-class manager like Pep Guardiola, who has a proven track record of developing young players like myself.
“I believe I can improve here, at a club whose ambitions match my own and I will give everything to be successful and win titles.”
Dias is City's third signing of the transfer window following deals for fellow centre-back Nathan Ake and winger Ferran Torres and ends a long wait to replace former captain Vincent Kompany, who left in May 2019.
The defender, who could make his debut at Leeds on Saturday, won the Portuguese title with Benfica after coming through the Portugal club’s academy and says both sides share the same desire to win trophies.
“The competition is different and here the opponents will be a level higher, but the mentality of the club is the same, it is win or win,” he said.
“I am a player that loves to win, a player for the team, for the fans, a player that loves this family spirit which I've felt a lot since I have been here.
“I am an aggressive player and I love what I do. I love the game and without getting into too much detail, because it is something you will see when I start playing, the most important thing to say is I love to win and that is the thing you most need to know about me.”
Dias has won 19 caps for Portugal and will join up with international team-mates Bernardo Silva and Joao Cancelo at the Etihad Stadium.
“We have been in touch, not just this year but for the last two years perhaps,” he added. “Obviously, the talks were amazing because you only hear amazing things about the club and about the environment.
“When the word came, I was just really happy to be able to join.”
Director of Football Txiki Begiristain believes they have got the right man having watched Dias's progress for the past two years.
“Ruben is a player we have admired for some time and we have been impressed with how he has progressed at Benfica,” Begiristain said.
“He has developed into a real leader and as a defender, he has all the attributes we are looking for. He is strong in the air, good in one vs one situations and technically he is very good, so we are confident he can thrive in our system.
“He is another talented addition to the squad and someone who is only going to get better working under Pep.”