The Blackhawks traded second- and seventh-round picks in next year's draft and a third-rounder in 2021 to the Canadiens for Shaw and a seventh-rounder in 2021.
Shaw was selected by Chicago in the fifth round in 2011 and spent his first five seasons with the Blackhawks, helping them win the Stanley Cup in 2013 and 2015. He was traded to the Canadiens in June 2016.
The 27-year-old Shaw had 19 goals and a career-high 28 assists in 63 games last season. He is signed through the 2021-22 season with an average annual value of $3.9 million.
Chicago has been active this offseason after missing the playoffs for the second straight year. It also acquired defensemen Olli Maatta and Calvin de Haan in a pair of trades.
The trade was announced Sunday.