The team with the worst record in the Eastern Conference defeated the team with the second-best record in the West.
Vancouver took the lead midway through the second period but it was all Montreal after that.
Blake Geoffrion, who made his Montreal debut Tuesday after being obtained in a trade with Nashville, scored his first goal of the season to tie it.
Montreal then scored three times in the third period, with Cole giving the Canadiens the lead 13 seconds into the stanza. P.K. Subban had a power-play goal 5:20 into the period and Cole added his 25th goal of the season.
Carey Price made 32 saves to get the victory. Roberto Luongo surrendered all four goals on 36 shots.