The Denmark international will join the Gunners as a replacement for Jorginho, who left the team last month to join Flamengo.
Arsenal's manager, Mikel Arteta, wanted to bring Norgaard to the team as a way to add experience to the squad.
With Thomas Partey also leaving the team, Norgaard's signing is seen as an important move to strengthen the midfield.
Brentford's new manager, Keith Andrews, has given Norgaard his best wishes as he leaves the club to join a bigger team.