Telles, 27, has put pen to paper on a four-year contract at Old Trafford.
Sources told ESPN on Sunday that United had agreed a fee around €15 million with Porto for Telles ahead of Monday's transfer deadline.
"To join a club with the prestige of Manchester United is a huge honour," Telles told United's official website. "You have to work hard to get to this moment in your career and now I am coming to this club, I can promise that I will give everything in my heart to be a success here.
"I won many trophies at FC Porto and I want to continue that at United. The manager has a clear plan and direction for this team and I can't wait to pull on the famous shirt."
United manager Ole Gunnar Solskjaer added: "First and foremost, I welcome Alex to United. He is a player we have been tracking for some time and his performances over the past few years are exactly what we are looking for. He is a fighter and a winner and will add real determination and competition to the squad.
"Alex has the qualities, both as a player and as a person, that we want here at Manchester United."
The defender ends his four-year spell with Porto where he won two league titles and scored 21 goals in 127 appearances for the club.
United have struggled defensively this season, conceding 11 goals in three Premier League games including the 6-1 defeat at the hands of Jose Mourinho's Tottenham.