The Jets were leading 3-2 after two periods, but Carolina put on the pressure in the third, firing 19 shots at Ondrej Pavelec in the game's final 20 minutes.
Eric Staal scored his second of the night to tie it up at 10:24 and Chad LaRose had the winner with 1:22 left.
"This one will stick in there for a long time," Noel said. "To me, the bottom line with this game is that if we don't make the playoffs, we don't deserve to make the playoffs. That's the bottom line. We don't deserve it.
"You can go down the road record all you want, you can go down all the other areas you want, but if we don't make it it's because we don't deserve to make it."
Winnipeg remains two back of the Washington Capitals for the eighth and final playoff spot in the Eastern Conference.
The Jets have 76 points on the season, tied with the Buffalo Sabres. However, Winnipeg has a game in hand.
"There's a price to pay to win games and there's a price to pay to get in the playoffs. And it's not a sometimes thing," said Noel.
Jussi Jokinen had a short-handed goal and added a pair of assists for Carolina (28-29-15). Cam Ward made 30 saves for the win.
Antti Miettinen had back-to-back goals for Winnipeg (34-29-8) in the second period, while Nik Antropov had his 13th goal of the season. Pavelec stopped 21 shots for the Jets.
The goals were Miettinen's first and second of the season and came in his 35th game.
"It's pretty irrelevant right now," Miettinen said of his new-found scoring touch.
He wasn't sure if the loss was due to his team's bad play or Carolina's strong effort.
"I don't know if it was because of them or because of us," said Miettinen. "I don't know if it was a lack of energy with too long of shifts or something, but the third period, I think, it was more about us going down than going up."
Winnipeg fans kept up their jeering chants of the visiting team's best player, this time yelling to Staal, "Jordan's better" in reference to his brother who plays for Pittsburgh.
"That was fun," said Staal, who scored his 20th and 21st goal of the season. "I mean, Jordan's good, but I don't know if he's better. It was fun to hear that, and it was fun to score a few and get the apple on the game winner.
"I think they probably regret the chants they were making as the game went on, started to hesitate and we stuck with it and were rewarded with a big win."
Carolina was coming off a 5-2 win over Minnesota on Saturday afternoon.
"It's a gutsy effort, back-to-back games coming in to a really tough building," Hurricanes head coach Kirk Muller said.
"This is the way our guys have been playing since Christmas. They go, give it everything they got. We had a short bench, had injuries there tonight and everything, but they've been finding a way to pick up points and I'm really proud of the group. Everybody's really stepped up."
The teams were tied 1-1 after the first period, but Winnipeg went up 3-2 after the second on Miettinen's second goal of the game.
LaRose scored the winner with a wristshot after Carolina won the faceoff in its own end and rushed down the ice. He has a goal in each of his past three games.
Staal's second goal came on a breakaway at 10:24 of the third period and tied the game 3-3.
Jokinen sent Staal a long pass that he picked up just before the blue-line. He finished the play with a shot that went over Pavelec's stick.
"The problem was is you couldn't stop the bleeding," Noel said. "That was the problem.
"That's sad. That's sad because we came so far. And to do this to each other and to disrespect each other like this was silly."
Carolina had a five-minute power play in the second period after Jets defenceman Zach Bogosian was tossed from the game for charging after he left his feet and knocked Andreas Nodi to the ice, who didn't return.
Bogosian had missed the previous five games with a lower-body injury.
The Jets killed off 3:43 of the penalty before the Hurricanes' Tuomo Ruutu shortened the advantage when he was sent to the box for interference.
Miettinen made it 3-2 with 1:06 left in the second on a quick passing play with Jets newcomer Spencer Machacek, who was called up from St. John's of the American Hockey League earlier in the day.
After crossing the blue-line, Miettinen sent a close pass to Machacek, who passed it back to him and Miettinen then fired a low shot by Ward.
Notes: Kane's assist extended his points streak to five games with two goals and three assists ... Tim Brent assisted on Staal's goal and now has two goals and two assists in his past four games ... This was the fifth of six meetings between the teams. Their last game is March 30 in Carolina. The Jets have won three of the five ... The Jets head out for a three-game road trip, starting with Pittsburgh on Tuesday, Washington on Friday and Nashville on Saturday ... Carolina heads home to host Florida on Wednesday.Suggest a correction