The Jets (40-26-12) trail the Los Angeles Kings for the second wild card from the Western Conference; they each have 92 points, but the Kings, who defeated the Colorado Avalanche on Saturday, are four ahead in the regulation/overtime wins column with four games remaining.
The franchise reached 40 wins for the first time since the 2006-07 season, when it was known as the Atlanta Thrashers.
Two goals from Lee Stempniak and three assists from Mathieu Perreault carried the Jets, who ended a two-game losing streak. Stempniak, an acquisition from the New York Rangers on March 1, has scored three goals in his past two games; he has five goals and three assists in 14 games with Winnipeg.
"It's huge to get these two points and move on to the next one," Stempniak said. "There is a lot on the line. Every game is so huge, and it will probably come down right to the wire, but we've got to give ourselves the best chance to keep staying in it and make the playoffs."
Vancouver (45-29-5) is second in the Pacific Division. The Canucks could have clinched a playoff berth with a win and other results.
"We've got to regroup now and try to clinch it at home," Canucks goaltender Eddie Lack said.
"They came out like it was a Stanley Cup Game 7," said Vancouver forward Alexandre Burrows, who scored twice.
Thirty-six seconds after Vancouver tied the game, Stempniak gave the lead back to Winnipeg.
Mark Scheifele made it 4-2 Winnipeg with 55.6 seconds left in the second period, but Burrows scored his 17th goal on a breakaway seven seconds later.
Stempniak's second goal with 9:36 left in the third period made it 5-3. Burrows scored his second of the game with 2:05 left with Lack off for a sixth skater. In the Canucks' final push, Chris Higgins hit the right post.
"There was no easy breathing in there for that one," Jets coach Paul Maurice said.
"We got the two points," said Pavelec, who is 7-2-0 starting nine of the past 10 games. "That's all that matters."
Slater scored for the third time in the past four games off a cross-slot pass from Jiri Tlusty. On the next shift, Frolik jammed a loose puck under Lack. Frolik's 18 goals are his most in a season since he scored 21 in 2009-10 with the Florida Panthers.
Lack, who has started 18 of the past 20 games with Ryan Miller (knee) out, faced 12 shots by the seven-minute mark of the first period.
"We knew they were going to come hard, and they did," Canucks coach Willie Desjardins said. "We gave up too much early. They're a desperate team, and they played hard."
Bryan Little's offensive-zone tripping penalty put Vancouver on its first power play. Alexander Edler's point shot ricocheted off the end boards into the crease, where Henrik Sedin stuffed it into the net. Eight minutes later, Daniel Sedin took Vrbata's pass from behind the Winnipeg net and slipped it past Pavelec.
Winnipeg's line of Stempniak, Perreault and center Adam Lowry combined for nine shots. They started to build another two-goal lead by scoring late in the first period when Stempniak tipped in Perreault's pass through the slot past Lack at 18:13.
In the final minute of the second period, Scheifele snuck a right-circle shot past Lack. But off the faceoff, Burrows beat Winnipeg defenseman Toby Enstrom and tucked a shot around Pavelec for his third goal in the past four games.
"I don't think that was the type of game we wanted to play, but that is what it evolved into," Canucks defenseman Kevin Bieksa said. "We could have been a little bit better defensively. We could have been a lot better on the walls. They're big, strong guys on the wall, and they won a lot of wall battles. Ultimately, I think that's where we lost the game."
The Canucks finished their road trip 2-2-0, losing the final two games. Lack has allowed 12 goals in the past three games.
"[To go] .500 against these teams is pretty good," Henrik Sedin said. "We've going to hold our heads high and come back home and play to win. If you would have put this group in this spot before the season started, I think we all would have liked to be here."
The Canucks play the Los Angeles Kings on Monday to begin a three-game homestand that will finish their regular-season schedule.
"As a player, that's what you play for," Burrows said. "You want to play in these playoff-[type] of games, and there is no better place than Rogers Arena to do it."
Desjardins said, "When you have a chance at home, that's all you can ask for. So we've got to make the most of it. We have an older group. They know that we need to do it, and we need to find some wins. There are no excuses; we've got to find [wins]."
The Winnipeg franchise has not reached the playoffs since 2007, but Stempniak said his teammates are adjusting well to the pressures of the race.
"It's hard," Stempniak said. "Your season is on the line. Every game right now is crucial to us being in the playoffs, and we're all well aware of it.
"The coaches have been preaching that hopefully we're going to be playing bigger games as we go along, so we've got to get used to playing them now, get used to the pressure, manage it, and at the same time have fun, which is an easy thing to say but hard to do at times."