The 25-year-old, who only made three starts in the Premier League, arrived in Germany on Thursday night before passing a medical and signing for the Bundesliga side for a fee believed to be around £12m.
The 16-time Dutch worldwide managed just 251 league minutes for Everton last season and didn't contribute a single goal or assist in that time.
He said the strength of Werder's desire to sign him made the club an obvious choice of destination.
Second Trump-Putin summit won't come until 2019: White House
He tweeted: "I'm very concerned that Russian Federation will be fighting very hard to have an impact on the upcoming Election". The indictment alleges that the Russians broke into Democratic email accounts and tried to penetrate state election systems.
"With his strategic sense, Davy is able to set a rhythm for the game, impresses with his creativity and is strong and aggressive at the same time".
"It was important to me to know exactly where the journey was going on the pitch and the ambitions of the club have convinced me", Klaassen told Bremen's website.
"After my exchange with (coach) Florian Kohfeldt I have a very good feeling that I can make an important contribution here with my abilities. I am really looking forward to that".