The 22-year-old defender looked destined for Lyon until United hijacked the deal and is new manager Erik ten Hag's first signing of the summer.
Malacia arrived at the Carrington training ground on Sunday and turned up on Monday to complete the second part, with a contract until 2026 being signed.
"It's an incredible feeling to have joined Manchester United," the new £12m signing said. "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."
Ten Hag has been a fan of Malacia since he was manager of Ajax where microphones picked up his comments about the young left-back.
Speaking to his coaches, Ten Hag said: "That kid at left-back is amazing".
He then instructed coach Michael Reiziger to take note of him.
Malacia recently revealed his ambition had always been to return to Old Trafford having played there with a Feyenoord youth side.
He also revealed he used to follow United thanks to another compatriot.
"I always followed Manchester United because Robin van Persie played there and he was a Feyenoord player. He was a lad who came from the academy at Feyenoord, too - just like me. That's why I got an extra feeling with this club.
"I talked to him [Van Persie] about the club and the city. About places to live. Also through my childhood friend Tahith Chong, with whom I played at Feyenoord, I have heard a lot.
"I'm only just coming to this level so I still have to show what I can do. But I want to reach the very highest and go for it - and I think I can do it, yes."
The defender made 136 career appearances for the Dutch side and has represented his country at every level from the Under-16s to the senior team, with five full international caps to his name.
Lisandro Martinez, Christian Eriksen and Frenkie de Jong are other players the boss hopes to secure, while Cristiano Ronaldo's future is up in the air.