Bremen announced the acquisition on their official Twitter account:
According to Phil Kirkbride of the Liverpool Echo, the Bundesliga side paid a fee in the region of £12 million for the player.
In a statement on their official website, Bremen boss Florian Kohfeldt said he was delighted to sign the Netherlands international.
"We have found our ideal midfield player," he said. "We have been watching him closely for a long time now, and I am thrilled that we have managed to bring him here. Davy is a creative, athletic and aggressive player who is able to control the pace of the game. He will be a focal point of our team."
Klaassen leaves Everton after one year, having failed to make an impact in the Premier League. He made just three starts in English football's top flight and played a total of 251 minutes.
There was excitement at Goodison Park when Klaassen arrived from Ajax last summer as part of a significant recruitment drive. The midfielder had impressed in the UEFA Europa League for the Dutch giants in 2016-17, captaining a young side to the final of the competition.
However, the Premier League never appeared a natural fit for the midfielder. While his off-the-ball movement was astute, he was laboured in possession and struggled to deal with the physicality of English football.
As Luke O'Farrell of ESPN noted, in fairness to Klaassen, he was part of a muddled transfer strategy from the Toffees:
Although German football is also intense, the Bundesliga should be more suitable for Klaassen, and he could become an important part of the Bremen side. At Ajax, he showed he can be effective in possession and dangerous when arriving late in the box.
In the campaign before his switch to Everton, he netted 16 goals and grabbed 10 assists in all competitions.
Bremen will hope Klaassen's arrival can give them a boost for the 2018-19 campaign, as they finished in a disappointing 11th position in the Bundesliga last season.