Luongo, who stopped 20 shots, turned aside David Desharnais, Max Pacioretty and Michael Cammalleri in the tiebreaker as the Canucks won their third in a row to begin a five-game trip. Vancouver, which trailed 3-0 in the second, has won eight of nine.
"After the third goal I just wanted to shut the door, and hopefully the boys would bail me out, and they did," Luongo said.
Sami Salo drew Vancouver even at 3 with a power-play goal with 4:40 left in the third. Salo drove a slap shot from the point past Price as Montreal's penalty killers allowed their first goal on home ice since Oct. 24.
Mason Raymond scored a short-handed goal in the second to begin the Canucks' comeback.
"That kind of got us going," Luongo said. "We've been a really good third-period team all year long and we knew that we could come back in the game. We got a huge goal early in the third, and it really gave us the momentum."
It was Raymond's first goal in three games since returning from a vertebrae compression fracture sustained in the Stanley Cup finals.
"I'm feeling pretty good," Raymond said. "I'm continuing to get better, unfortunately it takes games and more games to get really where you want to feel, but I'm very pleased with where I'm at considering where I was."
Hodgson, who deked before shooting over Carey Price on Vancouver's first shootout attempt, scored during 4-on-4 play 4:24 into the third to draw Vancouver within 3-2.
Price made 28 saves, and Raphael Diaz also scored for Montreal, which has lost six of seven (1-3-3).
"At least we're picking up points, but singles aren't helping us right now," said Price, 1-5 in shootouts this season.
The Canucks were without David Booth, who sprained his right MCL on a knee-on-knee hit with Colorado's Kevin Porter in Tuesday's 6-0 win over the Avalanche. Porter received a four-game suspension from the NHL earlier in the day. Booth is expected to miss four to six weeks.
Raymond got Vancouver back in the game with Montreal on the power play. He toe-dragged the puck before snapping a shot past Price to draw the Canucks to 3-1.
The Canadiens have allowed five short-handed goals.
Hodgson made it 3-2 with an unassisted effort for his sixth goal.
"It's disappointing but when you look at it, it's simple -- special teams," Canadiens coach Jacques Martin said. "They scored short-handed, 4-on-4, and on the power play. I think 5-on-5 we played well. It wasn't a lack of effort. It was one of the best teams in the league and we competed with them, but special teams made the difference."
Frederic St. Denis got his first goal in eight games with the Canadiens to open the scoring 4:38 in. His wrist shot from the point got past Luongo, who complained to no avail to the referee after Montreal's Tomas Plekanec jostled for position with defenseman Kevin Bieksa in front of the net.
Diaz made it 2-0 with 1:09 to go in the first on another shot from the point that found its way past Luongo through traffic in front.
Cole, who got an assist on Diaz's goal, beat Luongo through the pads with a low shot from the right side 2:20 into the second to make it 3-0.
Raymond's goal was the Cancuks' first short-handed tally of the season. ... Canadiens D Yannick Weber sat out in favor of St. Denis, who has alternated in and out of the lineup for eight games.