Zucker, 28, has been rumored to be part of several deals over the years and was reportedly headed to Pittsburgh last year, until Penguins winger Phil Kessel refused to waive his no-trade clause.
Zucker, who has 14 goals and 15 assists this season, had career bests of 33 goals and 31 assists in 2017-18. He has spent his entire nine-year career with Minnesota, which drafted him in the second round of the 2010 draft. He is in the second year of a five-year, $27.5 million contract.
"Jason Zucker is a top-six forward that we think will be a great fit for our group," Penguins GM Jim Rutherford said in a statement. "He's a proven goal scorer, and with three more years on his current contract, he has the potential to make an impact for us moving forward."
Galchenyuk has five goals and 12 assists in his first season in Pittsburgh, his third team in as many seasons. His best season was 2015-16, when he recorded 30 goals and 25 assists for the Canadiens.
Galchenyuk, 25, has a $4.9 million salary and is scheduled to be an unrestricted free agent at season's end.
Addison was a second-round pick for the Penguins in 2018 and has yet to play in the NHL.