Gurianov scored three times in a five-goal second period after Dallas fell behind 3-0 6:34 into the game.
"Boys just said, 'Shoot the puck,'" Gurianov said. "I'm glad I got a hat trick and I'm glad we won. It's the most important thing, moving forward and getting ready for the second round."
It is the sixth time an NHL team has trailed 3-0 in the first period then rallied to win a series-ending game.
"We talked in the first period," Gurianov said. "We came in the locker room and just forgot about the first period and just moved on and just started to play to our game."
[WATCH: All Stars vs. Flames highlights]
Miro Heiskanen had one goal and three assists, and Joe Pavelski had one goal and two assists for the Stars. Anton Khudobin made 38 saves.
Dallas, the No. 3 seed, will play the No. 2 seed Colorado Avalanche in the Western Conference Second Round. Game 1 is Saturday (8 p.m. ET; NBC, SN, CBC, TVAS).
"It just feels good to win," Pavelski said. "It feels good to end it tonight, earn an extra day or two and start preparing for the next round, because there's always another team waiting and another big challenge ahead of us. We'll enjoy it for a bit and understand where our game's at and go from there."
Video: DAL@CGY, Gm6: Gurianov records four-goal game
Johnny Gaudreau scored for the Flames, the No. 6 seed in the West who lost the final three games. Cam Talbot made 11 saves on 15 shots, and David Rittich made six saves on nine shots to end the second period.
"This is an unbelievable group of guys," Calgary captain Mark Giordano said. "We all care about each other, so it's really tough. A lot of teams say that, that they have a great group, but we really bonded this year and we really were enjoying the [hub city] as a group. It's a tough one right now because you go back game by game in the series and we had a lot of opportunity and we couldn't get it done, so tough to swallow right now."
Heiskanen got the Stars within 3-1 with a power-play goal at 9:36 of the first. It came 3:02 after Dallas called timeout following Calgary's third goal.
"Fortunately, we have good leadership on this team," Stars coach Bowness said. "They could all see what we could see. It was just a mess to start. There comes a point where, 'OK, that's enough. We need the next goal. We get the next goal and we're back in the game.'
"That's what happened. It takes a special player like Miro to step up. 'We need a goal? I'm going to step up here and show up.'"
Video: Enterprise Hat Trick: Gurianov scores four vs. Flames
Gurianov scored 59 seconds into the second period to make it 3-2, and again at 3:25 to tie it 3-3. Talbot, who entered the game with a 2.17 goals-against average and .934 save percentage in the postseason, was replaced by Rittich after the goal.
Radek Faksa put Dallas up 4-3 at 5:47, and Pavelski made it 5-3 at 7:22.
"It's tough. An empty feeling," Calgary forward Mikael Backlund said. "It's still just happened. Personally, I just feel empty. You wonder what happened, things just turned so quickly. We just couldn't stop the bleeding. And it happened so quick. It's hard right now."
Gurianov completed his first Stanley Cup Playoff hat trick on a 1-on-1 to put Dallas up 6-3 at 15:30 of the second. He scored on Talbot, who replaced Rittich to start the third period, at 9:02 for the 7-3 final.
The 23-year-old, playing in the NHL postseason for the first time, is the first Dallas/Minnesota North Stars player to score four goals in a playoff game, and the first with four goals in a playoff game since Jake Guentzel for the Pittsburgh Penguins against the Philadelphia Flyers on April 22, 2018.
"Give him credit. He battled back like the rest of the team," Bowness said. "But hockey is an amazing thing. It's an amazing game sometimes. When you're hot like Denis was all of a sudden, the puck is on your stick at the right time. Those nights don't happen very often, but we all know Denis is and is going to be a special player.
"He had one of those nights. Good for him."
Video: Clutch Performance: Pavelski tallies three points
Calgary scored three goals on its first eight shots.
Andrew Mangiapane scored at 3:42 to put Calgary up 1-0, and Gaudreau extended the lead to 2-0 with a power-play goal at 5:38. Rasmus Andersson made it 3-0 at 6:34.
"We just couldn't stop the bleeding in the second, put ourselves in a tough spot going to the third," Backlund said. "But yeah, very good first period, for sure our best first period this playoffs. It came at the right time. It's too bad we can't manage to follow it up. It's tough."
NOTES: Dallas' rally from down 3-0 was the first since Calgary defeated the Vancouver Canucks 7-4 in Game 6 of the 2015 Western Conference First Round. ... The Flames have never won any of 10 series after trailing 3-2. ... The Stars are 16-3 after taking a 3-2 series lead.
NHL.com staff writer Tim Campbell contributed to this report
Video: Gurianov scores four goals in Game 6 victory