The Islanders, who landed Vanek earlier this season in a trade with the Buffalo Sabres, received a conditional second-round draft pick in 2014 and prospect Sebastian Collberg from the Canadiens, who also got a conditional fifth-round pick.
The draft picks will be exchanged only if the Canadiens (34-22-7), who entered Wednesday with the third-best record in the Eastern Conference, reach the playoffs this season, a source told ESPN.com's Pierre LeBrun.
"When you think of a hockey mecca, Montreal is it," Vanek said on a conference call. "I played many times for Buffalo in Montreal, and it's so electric. I'm super excited."
Vanek has 21 goals and 32 assists in 60 games, and is now the Canadiens' leading scorer. He should provide some punch for Montreal, which ranks 20th in the league in scoring.
Vanek, who is in the final season of a $50 million, seven-year contract and will be an unrestricted free agent this summer, had told the Islanders he wasn't interested in signing a contract extension and was eager to test the free-agent market.
Vanek would not speculate on whether he would seek a new contract to stay in Montreal after this season.
"It's been a tough year for me and my family," he said. "I knew there was a chance we'd move again. Now it's over and I'm a Montreal Canadien and I can just focus on hockey and helping that team win a lot of games."
A former All-Star, Vanek was traded to the Islanders this past October in the deal that sent Matt Moulson and two draft picks to Buffalo. The 30-year-old Vanek has 271 goals and 270 assists in his nine-year career with the Sabres and Islanders. He was picked fifth overall by Buffalo in 2003.
Collberg, 20, has been playing with Frolunda of the Swedish Elite League this season. Montreal selected him in the second round of the 2012 draft.
ESPN.com's Pierre LeBrun and The Associated Press contributed to this report.