Montreal had 18 shots and made the most of very little offence against Henrik Lundqvist.
The Canadiens tied it late in the second on Max Pacioretty's goal and went ahead for good when Galchenyuk matched Pacioretty with his second of the season 1:48 into the third.
Raphael Diaz added an empty-net goal with 1:06 remaining for the Canadiens, who have outscored opponents 15-5 during their streak.
Defenceman Anton Stralman netted his second of the season in the second period, but it wasn't enough for the Rangers, who played without top-line forward Rick Nash (undisclosed injury) and had a five-game point streak snapped (4-0-1).
LIGHTNING 4, MAPLE LEAFS 2
Toronto goalie Ben Scrivens had his bid for a third consecutive shutout thwarted midway through the first period on Lecavalier's goal. The Maple Leafs, 7-3-0 on the road this year, got goals from Mikhail Grabovski and James van Riemsdyk.
JETS 2, SABRES 1
BUFFALO, N.Y. (AP) — Ondrej Pavelec made 29 saves to help Winnipeg snap a three-game skid with a win over Buffalo.
Jason Pominville scored for Buffalo, once again booed by its home crowd. The Sabres are 4-10-1 in their past 15 and have dropped consecutive home games for the first time since last February.
SENATORS 3, ISLANDERS 1
Colin Greening also scored for the Senators.
PREDATORS 4, RED WINGS 3, OT
NASHVILLE, Tenn. (AP) — Shea Weber scored 44 seconds into overtime to give Nashville a victory against slumping Detroit.
SHARKS 2, BLUES 1
The Sharks' skid was their longest since a 10-game slide in November 2005.
Joe Thornton also scored for San Jose, which began the season with a franchise-record seven successive wins.
St. Louis has dropped five consecutive home games. The Blues also had a three-game winning streak halted. They were coming off a 3-0 road trip with wins in Detroit, Calgary and Vancouver.
Kennedy scored the go-ahead goal at 13:02 on a shot from the left faceoff circle that eluded goalie Jake Allen, who had won the first three starts of his career.