The 19-year-old moved to Old Trafford from Tromso in his homeland in 2020 and went on to gain regular experience with the club's academy sides.
Hansen-Aaroen has scored two goals in eight Premier League 2 appearances for the Red Devils this season, while he has also featured in two UEFA Youth League fixtures away against Copenhagen and Galatasaray.
The Norway Under-21 international has decided to continue his development in Germany with Bundesliga outfit Bremen, penning a contract until the summer of 2028.
"Moving to Werder is the right step for me. I'm pleased that I've got the opportunity to play in the Bundesliga for such a big club," Hansen-Aaroen told Bremen's official website.
Bremen's head of professional football Clemens Fritz added: "Isak is a very exciting prospect who has a lot of potential. We're really happy that we've managed to convince him that Werder is the best place for his development.
"His contract would have expired in summer and a lot of suitors would have been vying for his signature. That's why it's even better that we've pulled off the transfer now."
Head coach Ole Werner continued: "Isak is a very skilful player. It's hard to get the ball off and has a good understanding of the game. He's got a lot going for him already, but he's got a lot of potential to get even better."
Hansen-Aaroen is the second player to depart Man United on a permanent deal in the winter window after Mateo Mejia, who signed for Sevilla a fortnight ago.