Daniel Alfredsson provided a goal and two assists while Jakob Silfverberg and Kyle Turris also scored for the Senators, who grabbed the advantage in their best-of-seven Eastern Conference quarterfinal series.
Craig Anderson turned aside 33-of-34 shots.
After Turris scored in third period to make it 4-1 for Ottawa, a series of fights broke out at center ice involving 10 players, each of whom were assessed fighting majors.
Two more brawls broke out later, resulting in a total of 236 penalty minutes, including 25 for Montreal's P.K. Subban and 19 for Ottawa's Jared Cowen.