Johnny Gaudreau scored the overtime winner at 15:09.
He collected a rebound and put a sharp-angled shot over Stars goalie Jake Oettinger, who made 61 saves in the game.
The Flames will face the Edmonton Oilers in the Western Conference's semifinal in the first post-season Battle of Alberta since 1991. The series opens Wednesday in Calgary.
Calgary's goalie Jacob Markstrom stopped 26 shots and had an assist in the win.
After reaching the Stanley Cup final in 2004 when Calgary fell in seven games to the Tampa Bay Lightning — 2005 was a lockout year — the Flames didn't win another playoff series until 2015 when Calgary was bounced in the second round by Anaheim.
Calgary (50-21-11) headed into this year's playoffs the higher seed atop the Pacific Division, while Dallas (46-30-6) earned the first wild-card spot in the conference.
Playoff success scant in recent years, the Flames were under considerable pressure to win the first Game 7 in 16 years in front of a sellout at the Saddledome, and back up a stellar regular season with some post-season progress.
Calgary outshot Dallas 52-23 over three periods Sunday, but the score was deadlocked heading into the first overtime period of the series.
Oettinger stopped Gaudreau on his doorstep near the 11-minute mark of the third, followed by Markstrom stoning Jacob Petersen on a breakaway.
The Flames drew even in the second period on Tkachuk's goal a second after a Dallas minor expired.
Markstrom head-manned the puck to Gaudreau, who dished to Tkachuk for a sharp-angled shot top corner at 8:44.
Namestnikov made it 2-1 for Dallas just 31 seconds after Toffoli had tied the game.
Namestnikov one-timed a wrist shot by Markstrom on a lateral feed from Joe Pavelski off the rush at 2:17.
Toffoli tipped in Oliver Kylington's shot from the boards just inside the blue-line at 1:46.
The Flames outshot Dallas 17-8 in the first period, but the visitors led 1-0 on Benn's goal 40 seconds after puck drop.
Tyler Seguin from the side of the net fed Benn in the mid-slot and the Stars' captain beat Markstrom stick side.
Calgary's top shutdown defenceman Chris Tanev was also scratched with an undisclosed injury.
Almost half of the 2022 edition of the Flames had never won a playoff round, while the Stars were deep in that experience having lost to the Lightning in the Cup final two years ago in Edmonton.
The series was a goaltending showcase.
On the back of a breakout performance by 23-year-old Oettinger and punishing defence around their crease, the Stars held a high-scoring offence to 10 even-strength goals in seven games.
Markstrom saw less rubber than Oettinger, but made the acrobatic, timely saves to give his team a chance to win in every game and held off the Stars when Dallas pressed early in the extra period.
The Stars boasted more recent Game 7 experience having edged the Colorado Avalanche in overtime two years ago to reach the conference final.
But Flames coach Darryl Sutter has coached an abundance of them with his 8-3 record now unmatched by any other NHL coach in wins.
Calgary opened the series with a 1-0 win and a 2-0 loss at home before earning a split at American Airlines Center.
The Flames then won 2-1 at home and lost 4-2 in Dallas to send series back to Calgary for the finale.
This report by The Canadian Press was first published May 15, 2022.
Donna Spencer, The Canadian Press