Suzuki scored shorthanded at 15:20 to make it 5-1, and Tyler Toffoli cut it to 5-2 with a 5-on-3 power-play goal at 17:04.
Montreal had 15 shots on goal in the third period.
"We had a good start, we did the right things," Beauvillier said. "They had a push late in the third period, but our goalie was really good for us. We were hard to play against, which is really our identity. We took care of all the details that have brought us success in recent games. It was really a game in our image."
Nelson scored his fourth of the game into an empty net with 19 seconds remaining for the 6-2 final.
"We've just got to find a way," Canadiens forward Jake Evans said. "I think there's some mental lapses, just some tougher plays, tougher reads that are costing us, and it seems like everything's going in the back of the net."
NOTES: Nelson is the first Islanders player to score four goals in a game since Kyle Okposo on Jan. 16, 2015. It was his third NHL hat trick and first since Feb. 10, 2018. … Montreal played without Jonathan Drouin; the forward is day to day after being struck in the head by a shot in the first period of a 3-0 win against the Detroit Red Wings on Tuesday. … Chara played his 1,616th NHL game and passed Larry Murphy for third on the all-time list for games played by a defenseman. Chara had an assist, his first point since rejoining the Islanders. … Parise had two assists, his first points with New York. … Their four goals in the second were the most the Islanders have scored in a period this season.