The six-foot-five, 215-pound winger has appeared in 480 regular season games over eight NHL seasons with the Winnipeg Jets and the Blue Jackets, scoring 204 goals and adding 184 assists.
Laine, from Tampere, Finland, was drafted second overall by the Jets in 2016.
Laine helped Finland win the 2016 world junior championship in Helsinki with seven goals and six assists in seven games, in addition to being named to the tournament's all-star team.
Coming the other way in the deal, the 24-year-old Harris had three goals, 11 assists and a minus-5 plus/minus over 56 games with Montreal last season. Harris has eight goals and 23 assists in his career over 131 games over three seasons with the Canadiens.
Laine signed a four-year, US$34.8 million contract extension with the Blue Jackets on July 22, 2022.