Jason Robertson scored, and Jake Oettinger made 29 saves for the Stars, the first wild card in the West.
"Big Game 6, (we) get to go home and play in front of our fans," Dallas captain Jamie Benn said. "We knew it was going to be a long series."
Robertson gave the Stars a 1-0 lead at 13:21 of the second period when the rebound off his initial shot bounced in off Flames defenseman Noah Hanifin.
It was Robertson's first career goal in the Stanley Cup Playoffs after he had 41 in 74 regular-season games.
"At this point of the year, I'm just excited that we're scoring goals," Robertson said. "I want to produce offense, and it's nice when I get to do that. But at the end of the day, it wasn't enough. There's not much to be happy about."
Backlund tied it 1-1 at 6:49 of the third period when he redirected a cross-crease pass from Mangiapane. He also scored in Calgary's 4-1 win in Game 4 on Monday, which ended a 20-game drought dating to the regular season (March 26).
"I felt great," Backlund said. "It's always a great feeling scoring at home, especially in the playoffs. There's nothing better than winning. I haven't scored much down the stretch here, so it felt really good to put the puck in the net, and such a big goal as well."
RELATED: [Complete Flames vs. Stars series coverage]
Mangiapane gave Calgary a 2-1 lead at 10:38 when he shot over Oettinger's glove from the slot.
"Yeah, it was big," Mangiapane said. "I was obviously a little happy there. Job's not done, obviously. We have to prepare for next game."
Trevor Lewis scored an empty-net goal with 58 seconds remaining for the 3-1 final.
Calgary outshot Dallas 16-5 in the third after having 16 shots through the first two periods.
"They took it to another level, and we didn't," Stars coach Rick Bowness said. "We didn't handle it very well. We stayed with them for two periods, obviously. They just cranked it up in the third period and we didn't handle the puck well to give us an opportunity to spend time in their zone. You want to handle the puck, you want to manage the puck, you want to spend some time in their zone, and we just didn't do that."
NOTES:The Flames are 7-3 in a best-of-7 series when winning Game 5 to break a 2-2 tie. Three of the wins were in six games. ... Calgary won twice in the regular season when trailing after 40 minutes, including once against Dallas. ... The Stars have been outshot in all four games this series 182-127). ... Dallas has yet to score a power-play goal on the road in this series (0-for-10).
