The matchup of Vezina Trophy favorites Lundqvist and Luongo fizzled as the New York netminder was gone 5:33 into the second period. He allowed five goals on 17 shots.
"You never know in this game," said Luongo, who made a season-high 39 saves. "Things happen."
Vancouver's first four goals resulted from Rangers turnovers, and the last was scored while the Canucks were on a two-man advantage.
Kyle Wellwood scored a breakaway 4:16 in and Burrows followed with another -- while the Canucks were short-handed -- to make it 2-0 at 9:38. A pass to Rangers defenseman Wade Redden bounced over his stick, creating the chance.
"The turning point in the game," said Rangers coach Tom Renney, who held a 10-minute postgame closed-door meeting. "I'm not going to suggest we were doing everything right at that point, but that was kind of the turning point and unfortunately it was really early."
The barrage got worse in the second period. Burrows took advantage of yet another Rangers giveaway and netted his sixth goal 1:23 in. Ryan Johnson pushed the lead to 4-0 58 seconds later.
Three Rangers penalties followed, giving the Canucks a pair of lengthy 5-on-3 advantages.
"That's our goal. Get on the forecheck and pressure their D men," Burrows said.
Vancouver appeared to score again, but Rangers defenseman Michal Rozsival swept the puck off the goal line just before it fully crossed. New York got a reprieve after the long video review, but the good fortune didn't last as the Canucks made it 5-1 just 20 seconds later at 5:33 when Pavol Demitra ripped in his fourth of the season and second in two games.
Lundqvist skated right to the bench after Renney summoned him with his finger. He was replaced by Steve Valiquette, who stopped all 15 shots he faced. Lundqvist, who started for the 14th time in 15 games, had gone 12 without allowing more than two goals.
"It was a good call," Lundqvist said. "It was a game where I had to be really sharp. There were a lot of situations where we got left alone out there. I have to be sharp. I wasn't."
Renney considered pulling him after the third and fourth goals, but decided to give Lundqvist a chance to fight back and the rest of the team the opportunity to play better in front of him.
"He was alone himself a couple of times," Renney said. "There was nobody to be seen. He is a great goalie, but not a great defenseman. He can't break it up, too."
Chris Drury scored a power-play goal, and Aaron Voros banked a shot in off of Luongo's back in the second period. Both goals trimmed the Rangers' deficit to three. Ryan Callahan added another man-advantage tally 7:45 into the third, but New York's winning streak ended at three.
The Rangers held a 17-5 shots advantage in the third, but couldn't pull off the comeback.
"For some reason, we kind of sat back," Burrows said. "We weren't as aggressive as we should have been, but Louie made some great saves again."
Daniel Sedin added an empty-net goal with 2.3 seconds left.
Luongo's best save came against Redden in the second. He was bailed out early in the third when Redden struck both posts with a shot while the Rangers were on a 4-minute power play. He smothered a shot with 5 minutes gone in the third and drew chants of "Louuuu" from the sizable contingent of Canucks fans at Madison Square Garden.
Luongo made his 14th straight start, but had a streak of allowing fewer than three goals snapped at six -- a stretch that included three of his NHL-leading five shutouts. He is due for a night off Thursday when the Canucks make the third stop of a four-game trip at Minnesota.
"We scored a lot of goals early, and especially against a goalie like that it's nice to get," Luongo said.
Wellwood sent the Canucks on the way to their fifth win in seven games (5-0-2) when he split defensemen Rozsival and Dmitri Kalinin and scored his eighth goal on 21 shots this season -- an NHL-best shooting percentage of 38.1 percent.
Burrows then scored the league-worst seventh short-handed goal against the Rangers. New York had allowed the same amount of goals while on the power play as it has while killing penalties, until Demitra scored.
Rangers C Scott Gomez (leg) missed his fourth straight game and probably won't play Saturday at Ottawa. ... LW Markus Naslund, who spent 12 seasons with Vancouver, assisted on Callahan's goal. ... Lundqvist earned an assist when Drury scored. ... D Shane O'Brien took five of the Canucks' six penalties. ... Canucks D Alex Edler (flu) returned from a two-game absence.