Anthony Peluso did a nice job controlling the puck in the Flames end before dropping it back to Postma at the blue-line. The defenceman, who used to play his junior hockey in Calgary, stepped into a hard slapshot that beat Karri Ramo.
It was the final game of the season for the Jets. Going back to its days in Atlanta, the franchise has missed the playoffs the past seven years.
The only thing left to be decided for the Flames will be whether they finish 26th or 27th. They are currently tied in points with the Islanders and each team has one game left. The Flames own the tie-breaker.
In his third-career start, and third in a row, Michael Hutchinson had 35 saves, including 19 in a spectacular third period to preserve the victory.
The 24-year-old's biggest stop came with four minutes left when he threw out his glove to stab a shot from Kenny Agostino, who had been sent in alone.
It was Calgary's final game at the Scotiabank Saddledome where they finish up 19-19-3. As the club saluted the fans at centre ice, the appreciative fans rose to their feet in a thundering ovation to acknowledge a rebuild year in which the Flames were more competitive than most expected.
Calgary has played in 49 one-goal games (25-18-7), which ties the NHL record set by Florida in 2010-11.
Down 3-1 halfway through the second, the Flames got a great chance to get back in the game when they got 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, this time Calgary took full advantage.
At 11:46, Monahan continued his impressive rookie season scoring his 22nd goal. It's the most goals for a Flames rookie in 24 years, dating back to 1989-90 when Paul Ranheim had 26 and Sergei Makarov had 24.
Twenty-four seconds later, Russell whipped a wrist shot past Hutchinson from 40 feet out to tie it, setting off a loud roar from the raucous sell-out crowd of 19,289.
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 defenceman TJ Brodie in the chest and caromed past unsuspecting Flames 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 face-off circle into the top corner.
Ramo had 22 stops to see his record fall to 17-14-4.
Notes: 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 did not return... Calgary C Mike Cammalleri (virus) did not play... Paul Maurice finishes the year 18-12-5 after taking over as Winnipeg's head coach on Jan. 12... 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 line-up against the Canucks on Sunday.