Matthew Tkachuk and Elias Lindholm scored, Johnny Gaudreau had two assists, and Jacob Markstrom made 24 saves for the Flames (12-4-5), who ended a four-game winning streak and a seven-game point streak (5-0-2).
"The outcome [stinks] right now, but at the end of the day we just got to find a way to get that one to overtime, or try to get two [goals] before, but we can't lose that one in regulation," Tkachuk said. "Those ones hurt, so we'll learn from it. Those are the games that don't sit too well with you."
Connor one-timed a pass from Wheeler over Markstrom's shoulder with 4:51 remaining in the third period to put Winnipeg up 3-2.
He is seventh in NHL scoring with 24 points (14 goals, 10 assists).
"He made a [heck] of a pass, under a stick backhand," Connor said. "As a shooter, you know where the net is, you know where you're going. Kind of that spatial awareness where you're going to shoot. Something like that you just try to get it off quick. High glove was open there."
The goal came after Hellebuyck made four saves on Brenden Dillon's minor penalty for tripping at 8:14. Hellebuyck allowed four goals on 14 shots before getting pulled in the second period of a 7-1 loss at the Minnesota Wild on Friday.
"'Helly' was great tonight," Connor said. "He had a couple flurries there, even on that power play. We're all on the bench hooting and hollering when that's going on. Everybody's laying it on the line there. It's good to see him get a win, too."
Andrew Copp scored into an empty net at 19:00 for the 4-2 final.
"I think in the third we were pushing hard for the third one, but it just didn't go our way tonight," Lindholm said. "Then they get one with four minutes left. I think all of us on the ice could have made a better read, but that's just the way it goes sometimes."
Tkachuk scored 26 seconds into the first period on a cross-crease pass from Gaudreau to put Calgary up 1-0. The Flames have scored first in an NHL-leading 17 games and are 12-2-3 when doing so.
