He made $750,000 this season and was set to become a restricted free agent July 1. He will make $1.025 million each of the next two seasons, a source told ESPN.com's Pierre LeBrun.
He scored the overtime winner in Montreal's first game of the playoffs against Tampa Bay. The 25-year-old dressed for 16 of Montreal's 17 playoff games and finished with three goals and four assists in a fourth-line role.
Weise had three goals and an assist in 17 games for Montreal after being acquired in a trade with Vancouver for defenseman Raphael Diaz.
Weise has 30 points in 179 career NHL games.
The Associated Press contributed to this report.