Montreal, who also sent future considerations to the Predators, will receive forward Dustin Boyd, goaltender Dan Ellis and future considerations in exchange for Kostitsyn, whose older brother Andrei plays for the Canadiens.
Boyd collected five points in 18 games last season for Nashville, while Ellis had a 15-13 record with a 2.69 goals-against average in a back-up role to Pekka Rinne.
Kostitsyn, Montreal's seventh round draft choice in 2005 and a Belarusian Olympian at the Vancouver Games, had 24 goals and 44 assists in 155 career games over parts of three seasons with the team.
But he was suspended in October by Montreal when he failed to report to the Hamilton Bulldogs of the American Hockey League after being demoted. He later asked for a trade.
During Montreal's unlikely run to the Eastern Conference finals last season, Kostitsyn was also told by the Canadiens that he was not welcome at one of the team's morning skates because of his work habits.
(Writing by Frank Pingue in Toronto; Editing by Ian Ransom)