The versatile Swiss 19-year-old is mainly a centre-back but is capable of playing anywhere across the back four.
He made his first-team debut for the Gunners at right-back in a 6-1 Capital One Cup win over Coventry last September and then made an appearance in the Champions League, coming off the bench in a 2-1 loss at Olympiacos.
Angha has also played for Switzerland at several age-group levels and will now look to further his career in Germany.
Nurnberg sports director Martin Bader said: "Martin Angha has been brought to a top level of football at Arsenal over the last few years.
"He has shown his potential in London and will make the next steps in his development at FCN. We're delighted to have signed such an interesting, young player.
"We've had our eye on him for a long time, he matches our idea of a player who could flourish."
Angha added: "I had a great time in London and thank Arsenal for everything they've done. Now I'm looking forward to the challenge of playing for FCN."