Cole tallied 29 seconds into the contest and then produced two more goals in the span of 45 seconds, the last one coming on the power play.
The Senators lost their third in a row and saw their playoff position further threatened. Ottawa is currently seventh in the Eastern Conference, but has only a two-point lead over both Washington and Buffalo.
Petteri Nokelainen and Lars Eller scored for the Canadiens, who had lost three in a row. Carey Price picked up the victory with 37 saves.
Craig Anderson surrendered three goals on 23 shots before being replaced by Ben Bishop, who gave up two more scores on nine shots.