The 21-year-old is the seventh permanent signing made by Forest since the Al Hasawi family's takeover of the npower Championship club, in addition to the loan signings of Daniel Ayala and Sam Hutchinson.
Lansbury has spent loan spells with Norwich and West Ham in the last two seasons, helping both win promotion to the Barclays Premier League.
The England Under-21 international has been unable to break into Arsene Wenger's midfield at the Emirates Stadium and has now opted for a permanent switch away.
Forest chairman Omar Al Hasawi told the club's official website: "Henri joins us as a young player with a rich pedigree and fits the bill perfectly as the kind of talented player that our manager Sean O'Driscoll is intent on bringing into the club."
Source: PA
Source: PA