Paul Postma scored the winning goal, and Hutchinson made 35 saves, including 18 in a perfect third period, as the Jets hung on for a 5-3 win on Friday night.
"Especially with where I started at the start of the year, I didn't think it would be possible to be in the NHL," said Hutchinson, who played in Winnipeg's final three games and went 2-1 with a 1.64 goals-against average and .943 save percentage.
Postma broke a 3-3 tie at 9:16 of the third, ripping a slap shot past Karri Ramo from the blue line.
"There was a lot of traffic in front of the net, and I just wanted to get it through," Postma said. "I took a shot, kept it low and it found a way into the back of the net."
It then fell upon Hutchinson to preserve the lead as the Flames buzzed around the Jets end, outshooting Winnipeg 14-1 before Blake Wheeler put the game away with an empty-net goal.
"I definitely had a lot of work in the third period, but at the same time, our guys played hard in front and blocked a lot of shots," the 24-year-old Hutchison said. "When your guys are working that hard in front of you, it gives you that added incentive to bear down and make those saves."
One of Hutchinson's biggest stops came with four minutes left when he threw out his glove to stab a shot from Kenny Agostino, who came in alone.
"Hutch played so well for us these last three games," Postma said. "He's got a bright future ahead of him, and we're looking forward to playing with him."
It was the final game of the season for the Jets, who failed to make the playoffs for the seventh straight season -- dating to the team's days in Atlanta.
"We didn't get the start that we wanted, but by the middle of the second period after those two power-play goals, it seemed like the momentum shifted and we had a great third period," Flames coach Bob Hartley said. "The boys were on a mission, they wanted that win for the fans but unfortunately we couldn't get it done."
It was Calgary's final game at the Saddledome, where the Flames went 19-19-3.
Calgary has played in 49 one-goal games (25-18-7), tying the NHL record set by Florida in the 2010-11 season.
"We've really tried to establish a work ethic and an identity of being one of the hardest-working teams in the league," Flames center Matt Stajan said. "We have a lot of work ahead of us. No one is satisfied in here."
Down 3-1 halfway through the second, the Flames got a great chance to get back in the game when they received a two-man advantage for 1:31. Mark Stuart put the Jets down two men when he caught Hudler in the face with a high stick.
Although just 1-for-9 with a two-man advantage coming into the game, Calgary took full advantage this time.
At 11:46, Monahan continued his impressive season by scoring his 22nd goal. It is the most goals for a Flames rookie since the 1989-90 season when Paul Ranheim had 26 and Sergei Makarov had 24.
Just 24 seconds later, Russell whipped a wrist shot past Hutchinson from 40 feet out to tie it.
The Flames cut into a 2-0 deficit on Hudler's redirection 1:04 into the second period.
Winnipeg restored its two-goal cushion at 8:48 when Klingberg curled out from behind the net and wrapped in his first NHL goal.
The Jets scored the only two goals of the first period.
The opening goal at 7:23 came off a bad bounce. Frolik's rising shot was going well wide of the net when it hit Calgary defenseman TJ Brodie in the chest and caromed past unsuspecting goalie Karri Ramo.
The second goal was much nicer. Bryan Little left a nifty drop pass for Kane, who ripped a wrist shot from the top of the circle into the top corner.
"I never ever put a lot of stock into the last games when you're out, except if they quit on each other, if they just say `It's fine,'" Winnipeg coach Paul Maurice said. "They didn't quit and there was nothing on the line."
Ramo had 22 saves but he fell to 17-14-4.
Rookie Jets D Jacob Trouba left the game halfway through the first period after a shot from Paul Byron hit his own stick blade and struck him in the face. He didn't return. ... Calgary C Mike Cammalleri (virus) didn't play. ... Maurice went 18-12-5 after taking over as Winnipeg coach on Jan. 12. ... The Flames signed Hobey Baker winner Johnny Gaudreau and his Boston College linemate Bill Arnold. Both will join the team for practice Saturday in Vancouver and will be in the lineup against the Canucks on Sunday.