Franzen took a cross-ice pass off the boards from Wings captain Nicklas Lidstrom, cruised into the high slot and wired a shot by Miikka Kiprusoff at 4:23 of the second extra period.
Detroit, the top-seeded team in the Western Conference, will meet either the sixth-seeded Dallas Stars, who face the Vancouver Canucks in Game 7 on Monday, or No. 5 San Jose in the second round.
It was the Wings' first victory at the Pengrowth Saddledome since April 3, 2006.
The Flames' loss capped a difficult day for the team as the organization was fined $100,000 and head coach Jim Playfair an additional $25,000 for their players' actions in a penalty-filled finish to Saturday's game in Detroit.
Backup goaltender Jamie McLennan was also suspended for five games.
Robert Lang scored Detroit's first goal and captain Jarome Iginla had Calgary's lone goal.
Detroit outshot Calgary 55-21 and dominated the faceoff circle Sunday.
Kiprusoff was outstanding in the series facing 255 shots, but he could not carry the Flames on his back to the second round of the post-season.
Lang was dinged for hooking at 11:43 of the extra period. Flames forward David Moss had a chance to end it on the ensuing power play, but his shot deflected high off the helmet of a sprawling Hasek.
Calgary forward Craig Conroy took a double-minor for high-sticking Tomas Holstrom at 6:44 of the extra period, but Kiprusoff was outstanding as Detroit's shots on goal crept over 50.
Kiprusoff shut down Henrik Zetterberg on two excellent chances and held the Flames in it even though they were outshot 17-5 in the period.
Detroit's Todd Bertuzzi hauled down Stephane Yelle for a holding penalty 32 seconds into overtime, but the Red Wings limited the Flames to two shots during the power play.
Detroit winger Daniel Cleary hit the post to Kiprusoff's left with 1:17 left in regulation.
Iginla scored his second of the series to give the Flames a 1-0 lead at 3:09 of the second period, but Lang evened up with his first of the post-season at 16:54.
Lang caught the Flames flat-footed on their own blue-line and fired the puck between the pads of Kiprusoff to tie the game.
After losing every draw at each end of the ice in the first period, the Flames finally won an offensive draw early in the second and it led to Iginla's goal.
Daymond Langkow fed Iginla in the high slot from behind the net and Iginla fired the puck past Hasek's outstretched pad.
Notes - Detroit was scoreless on five power-play chances while Calgary went 0-for-4 . . . Flames defenceman Mark Giordano caught a puck in the face off a Mathieu Schneider shot in the first period, but returned to the game . . . Flames defenceman Robyn Regehr (knee) skated Sunday morning, but was a scratch for the fourth straight game . . . Defenceman Brent Lebda was not in Detroit's lineup. His leg twisted awkwardly into the boards while checking Daymond Langkow Saturday. Kyle Quincey was Lebda's replacement . . . After losing two games at the Saddledome last week, the Wings switched up their stall assignments in the visitors' dressing room Sunday.