Rick DiPietro made 28 saves in goal for the Islanders (25-21-6), who improved to 4-0-2 in their last six games.
Radek Bonk and Mark Streit scored for the Canadiens (28-19-6), who have only six wins in their last 17 games.
Montreal's big scorers continued their collective slump as the third and fourth lines accounted for both of the Canadiens' goals. The first line of Saku Koivu, Chris Higgins and Michael Ryder has combined for two goals and three assists in the last eight games, while Alex Kovalev has a goal and an assist over the same span.
Cristobal Huet made 23 saves for the Canadiens in losing his second straight start and his fourth in his last six.
Montreal plays again Sunday afternoon in the second leg of the traditional pair of matinee games on Super Bowl weekend, hosting Sidney Crosby and the Pittsburgh Penguins in a re-match of Thursday's hotly contested 5-4 shootout win for the Penguins in Pittsburgh.
New York Islanders defenceman Radek Martinek left the game on a stretcher after suffering an apparent left ankle injury at 12:16 of the first period, falling awkwardly after taking a hit from Canadiens rookie Guillaume Latendresse.
With the game tied 2-2 in the third, the Islanders took advantage of some sloppy play by Montreal in its own end to score twice in a span of 1:45 to take the lead.
Hunter scored to go-ahead goal at 11:26 of the third when he pounced on a failed Canadiens clearing attempt, wheeled and fired past Huet for his second of the game and 14th of the season.
Satan scored his second of the game and 17th of the year at 13:11, jumping on a Sheldon Souray turnover and firing a wrist shot past Huet to make it 4-2 Isles.
The Islanders opened the scoring less than two minutes into the game when Canadiens defenceman Craig Rivet got caught up ice to allow a three-on-one break, with Satan finishing a give-and-go with Radek Martinek at 1:44 of the first.
Montreal evened the score at 10:45 when Bonk scored on a wrist shot from the slot for his ninth of the year and his third in two games.
The Canadiens took a 2-1 lead after some good work by the fourth line deep in the Islanders' zone, with Streit scoring his fifth on a nice deke between DiPietro's legs at 17:13 of the second period.
Canadiens winger Alexander Perezhogin - a healthy scratch the previous two games - took a holding penalty late in the second and it was costly as Hunter tied the game 2-2 at 1:35 of the third, three seconds before the end of Perezhogin's penalty.
Notes: Canadiens C Tomas Plekanec was awarded the Molson Cup as the team's player of the month for January. Plekanec had a team-high seven goals and 12 points in 12 games last month.