The 21-year-old has moved to the City Ground on a five-year deal that will cost Forest £15m.
Elanga had been with Manchester United since the age of 12 and made 55 appearances for the first team, scoring four goals.
The Sweden international is Nottingham Forest's second summer signing after Ola Aina joined on a free transfer from Torino.
"It's a proud moment, not only for me, but for my family as well," Elanga said. "It's the perfect next step for me in my career.
"It's a big step and I'm really looking forward to playing in front of the Forest fans at the City Ground. I had interest from elsewhere, but for me, Nottingham Forest feels like the perfect place."
He added: "I've been here with United and from the minute I stepped out for the warm-up, I could hear them from the changing room. It's a special place but being here with Forest, I feel like it will be even more special.
"I'm really ready for the challenge and I just want to embrace everything and get to work."
Ross Wilson, Forest's chief football officer, said: "Anthony had a number of excellent options in front of him to choose from so we are naturally delighted that from those he has chosen to become a Nottingham Forest player.
"From the first meeting we had, he was so committed to all aspects of how he can continue to develop working with Steve [Cooper] on the pitch, whilst he was also excited and embracing of the ambitions that Mr Marinakis has for the football club."
Elanga has 12 caps and three goals for Sweden.
