Telles agreed a four-year contract, with the option to extend by a further year, just 24 hours after United were thrashed 6-1 by Tottenham.
The 27-year-old is set to replace Luke Shaw at left-back as United manager Ole Gunnar Solskjaer tries to plug the holes in his leaky defence.
"To join a club with the prestige of Manchester United is a huge honour. 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," Telles told United's website.
"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."
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."
Telles, who won league titles among a number of honours at both Porto and Turkish side Galatasaray, is United's second major signing since the end of last season after the arrival of Ajax midfielder Donny van de Beek.
United followed the Telles swoop by announcing they had reached an agreement to sign teenage winger Amad Diallo from Serie A side Atalanta.
The 18-year-old Ivorian has been on the radar of United scouts since 2016, when he was spotted playing at Under-15 level.
Diallo scored on his Serie A debut in the first of three first-team appearances last season, after rising through the youth ranks at Atalanta.
He will join United in January, subject to medical, personal terms and a work permit, with the initial fee reported to be £20 million.
With just hours left before the transfer window closed on Monday, United were also close to signing former Paris Saint-Germain striker Edinson Cavani on a free transfer.
