Gustav Nyquist pegged one back for Winnipeg on a rebound at 6:14, but when Calgary needed a stop, they got one from Wolf, his best of the frame coming at the end of an extended spell of defensive play when he coolly turned aside a Mark Scheifele effort after the Jets star snuck down the left side and drove the net.
Winnipeg forced extra time with a pair of third-period markers, one from in tight by Alex Iafallo, while the equalizer caromed off Parker Ford with 7:03 left in regulation time.
Wolf was forced into a pair of big stops in the extra session off Jets blueliner Neal Pionk, the second coming after the Winnipeg defender spun into the slot with the puck before firing it at the Calgary netminder's blocker hand.
Justin Kirkland and Rasmus Andersson tallied for the Flames in the shootout - Kirkland with his signature snapshot, Andersson a five-hole tuck that slipped through Hellebuyck's wickets.
But Tanner Pearson's goal in the seventh round of the penalty-shot session stood up as the decider.