"It feels great. I've always wanted to play in the Premier League, so it's a very exciting opportunity for me," Sargent said in a statement.
"I think the Premier League is the best league in the world and it's going to push me and challenge me to be a better player. 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 at Norwich. I'm very excited to get started."
Canaries manager Daniel Farke said Sargent has "lots of potential."
"He's a very good striking option for us but is someone who can also play on the wing," Farke said. "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 think we have a player in Josh who can contribute and develop into a top goalscorer on Premier League level."
Sargent, 21, had scored 15 goals in 83 appearances for Werder since making his senior debut in December 2018.
Bremen sporting director Frank Baumann said on Monday it was "disappointing" that the American had asked not to play in the German Cup game on Saturday, after the transfer deal had been reached. Bremen lost the game. He added: "We wish Josh all the best for the future."
Other teams expressed interest, but Baumann said they fell short of Bremen's target, which had been reported to be around €10 million.
For the U.S. senior side, Sargent has made 16 appearances and registered five goals along with one assist.