The 22-year-old joins from Feyenoord and has signed a four-year-deal with the option of another year. The deal is set to cost United £13 million plus add-ons.
He is the first signing under new manager Erik ten Hag.
Malacia said: "It's an incredible feeling to have joined Manchester United. This is a new chapter for me, a new league with new team-mates and a tremendous manager [Erik ten Hag] leading us.
"I know from playing against his teams in the Eredivisie, the qualities that he has and what he demands of his players.
"Whilst I know that I am still young and will continue to develop, I can promise the United fans that I will leave everything on the pitch every time I pull on the red shirt. I'll always be thankful to Feyenoord for all they have given to me and my family.
"None of this would be possible if it wasn't for them. Now I'm ready to focus on the future with United, and help my new club achieve success."
Malacia looked set to sign for Lyon after the French side agreed a fee with Feyenoord only for United to make a late swoop. He will compete for a place at left-back with Luke Shaw. Sources have told ESPN that the club are now willing to listen to offers for Brazilian full-back Alex Telles.
Malacia arrived in Manchester on Sunday and was back at Carrington on Monday morning to undergo the second part of his medical.
He has been capped five times by Netherlands and leaves Feyenoord after playing more than 100 games for his hometown club.
John Murtough, Manchester United football director, said: "Tyrell is an exciting, dynamic young footballer with excellent experience for his age, including five seasons in the Eredivisie, a European final and full international honours for the Netherlands.
"We look forward to seeing him continue to develop under the guidance of Erik ten Hag and his coaching team in the years ahead.
"Everyone at Manchester United welcomes Tyrell to the club."