The 21-year-old is the second Werder Bremen player to join Norwich City this summer following the arrival of Milot Rashica earlier in the transfer window.
Sargent moves to Carrow Road on a four-year contract.
Sargent scored five goals in 32 Bundesliga appearances last season but Werder Bremen suffered relegation to the second tier of German football.
He has won 16 caps for the U.S national team and has scored two goals.
Speaking to canaries.co.uk, Sargent said: "I've always wanted to play in the Premier League, so it's a very exciting opportunity for me.
"I heard it's a good group of guys and I talked with the head coach already and I really like what they're trying to do here.
"I like to think my main position is a number nine, but I am also very versatile. No matter where I play, everyone can expect that I will give my all, my 100 per cent, and hopefully score some goals."
Norwich manager Daniel Farke added: "We're really happy to have Josh with us. He's at a young age with lots of potential.
"He's a very good striking option for us but is someone who can also play on the wing. He has pace and is also good with his head. We know he is always there with a big workload and has a fantastic character.
"We know we have to be a bit more creative in the transfer market. We think we have a player in Josh who can contribute and develop into a top goalscorer on Premier League level. We see his potential and are delighted to have him with us."
Norwich will mark their return to the Premier League with a home game against Liverpool on Saturday.
