The Swedish international started his career with Manchester United before joining Forest and becoming a key player in their rise to European football last season.
Elanga registered an impressive six goals and 11 assists as Forest finished seventh in the Premier League.
He arrives at Newcastle looking to make the right wing position his own, with the Magpies struggling for depth in that area of the pitch in recent times.
Newcastle manager Eddie Howe to the club's website: "I'm delighted to welcome Anthony to Newcastle United. He has been a key target for us so I'm delighted to secure him at this early part of pre-season.
"He is an exciting talent with attributes that make him a unique attacking threat. His pace, energy and ability to create and score goals will strengthen us and complement the way we want to play.
"Anthony is hungry to develop even further with us and to achieve success here, and we’re all looking forward to working with him."
Elanga said: "I'm excited, I'm really happy, but most importantly, I'm ready. I'm ready to put on this black and white top to really fight for this team and give everything for the passionate fanbase that lives and breathes football.
"I had an amazing two years at Nottingham Forest, they really helped me to become the player I am today, but I'm really happy to be here now.
"The club won a trophy last season and are building something unique, something special that I really want to be part of.
"I really want to buy into the culture, into the DNA of this club. I'm really happy to be here and I can't wait to get started."
Elanga featured 82 times in all competitions for Forest and before that played 55 games for Manchester United - also racking up 22 appearances for the Swedish national team.
His first competitive outing in Black and White could come in their Premier League opener against Aston Villa.