Anthony Peluso controlled the puck in the Flames end before dropping it back to Postma at the blue line. The defenseman, who played junior hockey in Calgary, fired a hard slap shot that beat Karri Ramo.
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.
The only thing left to be decided for the Flames will be whether they finish 26th or 27th in the 30-team NHL. They are currently tied with the New York Islanders, and each team has one game left.
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 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.
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, setting off a loud roar from the 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 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.
Ramo had 22 saves but he fell 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 didn't return. ... Calgary C Mike Cammalleri (virus) didn't play. ... Paul 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.