Forsberg dropped his stick in the offensive zone, retrieved it and took a cross-crease pass from Mikael Granlund for his third game-winning goal.
"I wouldn't call it a set play but it worked out," Forsberg said. "It has been done before."
Luke Kunin had a goal and an assist on his 24th birthday, and Juuse Saros made 20 saves for Nashville (13-10-1), including a pad save on Artturi Lehkonen late in the third period. The Canadiens forward had a breakaway at 18:20 and took their second shot of the period.
"It wasn't the prettiest game by any means, but we capitalized on some timely chances and Juuse kept us in it and we found a way to win," Kunin said. "So we'll take the two points and it was a good job by everyone."
Cole Caufield and Tyler Toffoli had two assists each, and Jake Allen made 40 saves for Montreal (6-17-3), which got its first point in three games since general manager Marc Bergevin was fired on Sunday.
"Overtime can go either way, both teams had chances," Canadiens forward Nick Suzuki said. "Collectively as a group, I thought we're heading in the right direction in the last few games. We've been playing a lot better and the wins will come for us."