The 25-year-old has signed a five-year deal and will provide the back-up to number one Tim Krul and deputy Steve Harper.
Elliot said: "I am looking forward to the challenge of joining a club like Newcastle United.
"This is a huge club and a huge opportunity for me to learn from two of the best keepers in the division."
Elliot became Alan Pardew's seventh summer signing, and the former Addicks boss was delighted to have got his man.
He said: "The goalkeeping position is one we felt we needed to strengthen. Rob is an excellent keeper who I have worked with before, and I know he will be a good signing for the club."
Newcastle and Charlton had earlier found themselves at odds over the move with the League One side unhappy at the Premier League outfit's pursuit of the player.