Morrow played 38 games for the Canadiens this season, scoring five goals and collecting 11 points.
The 25-year-old rearguard also posted a minus-11 differential, and averaged 16:39 of ice time per outing.
A former first-round pick of the Pittsburgh Penguins in 2011, Morrow has 103 games of NHL experience with Boston and Montreal.
Following the trade, the Canadiens now have 10 picks in the 2018 NHL Entry Draft coming up in June - one in the first round, four in the second, one in the third, one in the fourth, and three in the fifth.
Ten draft picks is the most the Canadiens have had in a single draft since 2003, when they had 11.