Galchenyuk, who was an unrestricted free agent, produced three goals and seven points in 14 games with the Minnesota Wild this past season.
The 26-year-old began the campaign with the Pittsburgh Penguins, collecting five goals and 17 points in 45 contests before they traded him to Minnesota in the Jason Zucker swap.
The Senators will be Galchenyuk's fourth team in three seasons. He notched 19 goals and 41 points in 72 games for the Arizona Coyotes in 2018-19 after spending five-plus campaigns with the Montreal Canadiens.
Galchenyuk is coming off a three-year pact he signed with Montreal in July 2017, carrying a $4.9-million cap hit.
The Canadiens drafted him third overall in 2012.