Elmander, 32, cannot count on regular first-team action with the Turkish giants and will instead attempt to earn a place in Chris Hughton's attack.
Galatasaray technical director Bulent Tulun told sportexpressen.se, the sports website of Swedish daily newspaper Expressen: "I can confirm Johan is moving to England.
"We have agreed with Johan, his new club Norwich and his agent Anders (Wallsten). We wish Johan the best of luck in the future."
The Sweden international left the Reebok for Gala in July 2011 after three seasons in the Premier League. He scored 22 goals in 108 appearances in all competitions.
Last season, Elmander only netted four times in 16 Super Lig outings as Galatasaray retained the domestic title.