The Capitals traded second-round picks in 2017 and 2018 to the Canadiens for the 27-year-old Eller, who had 13 goals and 13 assists in 79 games last season.
Eller has two seasons left on his contract at a salary-cap hit of $3.5 million. In 442 NHL games, the Dane has 73 goals and 83 assists.
In addition to trading Eller, the Canadiens acquired forward Andrew Shaw from the Chicago Blackhawks for two second-round picks in Friday night's draft. Montreal sent the 39th and 45th picks to cap-strapped Chicago for the pending restricted free agent.
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