Sedin took a pass from behind the net from his brother Daniel and blasted a shot past Dallas goaltender Marty Turco.
The victory came after the Canucks blew a 4-2, third-period lead. Vancouver was forced to play the sixth longest playoff game in Stanley Cup history to take a 1-0 lead in the best-of-seven series.
"It's a big relief to get off the ice," said Henrik Sedin. "To lose the game would have been tough."
Vancouver players streamed off the bench to congratulate Sedin. The exhausted Stars, who had outshot Vancouver 76-56 and dominated most of the overtime, dragged themselves from the ice.
"I'd be lying if I said we weren't disappointed to not get the win," said Brenden Morrow, who had a goal and assist for Dallas. "You have to think about it, park it, forget about it and move on.
"We're not hanging our heads. It's a long series. It's going to take four wins and we're not the type of team that's going to quit after losing one hockey game."
Game 2 of the best-of-seven series goes Friday.
Canuck goaltender Roberto Luongo made 72 saves in his first NHL playoff game, one less than Kelly Hrudey's playoff record.
"I think everybody was exhausted," said Luongo. "It's a hard game to play.
"Guys were battling the whole time and trying not to make mistakes."
Luongo laughed when asked how frustrating it was to see a two-goal lead evaporate.
"It's not frustration because we won," he said. "It's such a long time ago I don't even remember."
The overtime lasted 78 minutes, six seconds. The longest overtime game in NHL history was 116:30 when Detroit beat the Montreal Maroons 1-0 back on March 24, 1936.
"It was a strange game," said Henrik Sedin. "Personally I felt better the longer the game went. I got my legs going in the fifth and sixth period."
It was a rough night for Turco, who is trying to redeem himself after Dallas had two first-round exits from the playoffs.
"It's really disappointing right now," he said. "We had a big chance to win a game tonight.
"It's frustrating but I'm sure we'll turn it into a positive, too. It's tough right now to think like that but we carried the play. We just have to find a way to get more pucks ... and sustain the pressure down there even more."
The Stars notched goals five minutes apart in the third period to force the overtime.
Antti Miettinen sparked the comeback by scoring at 8:31, and then assisted on Ladislav Nagy's tying goal at 13:46.
Trevor Daley had the other Stars goal.
Markus Naslund had a goal and an assist for Vancouver. Bryan Smolinski, Daniel Sedin and Mattias Ohlund, on the power play, also scored for the Canucks who missed last year's playoffs.
The game started at 7:11 p.m. local time and didn't end until 12:35 a.m.
Many in the sellout crowd of 18,630 had left before the longest game in Canuck history had ended.
The 76 shots given up by the Canucks was a franchise record. That exceeded the 54 the New York Rangers managed against Vancouver in Game 1 of the 1994 Stanley Cup final.
Vancouver's 56 shots on goal also broke the franchise record of 47 set against Toronto on May 24, 1994.
Dallas played 80 minutes, 48 seconds of overtime in a 4-3 loss to Anaheim on April 24, 2003. In that game the Stars had 63 shots on goal.
Canuck veteran Trevor Linden, who celebrated his 37th birthday Wednesday, said a loss would have been devastating for Vancouver.
"It would have been very difficult to lose this game," said Linden. "To be up two and let that go, to lose would have been a bitter pill to swallow.
"It's nice we got the win and can move on."
Vancouver paid a toll for the victory. Grinding winger Matt Cooke left the game after the second period with what the team said was a groin injury.
Forward Alex Burrows was clobbered into the boards on a hit from behind by Dallas's Mike Ribeiro in the first overtime. A groggy Burrows barely managed to stagger to the bench and didn't return for the rest of the game.
Centre Ryan Kesler, out since January after hip surgery, was a surprise addition to the Canuck lineup.
Kesler had surgery Jan. 29 to repair a torn labrum, the cartilage that lines the socket of the hip. He missed the rest of the regular season but began skating with the Canucks this week.
The Canucks finished first in the Northwest Division - claiming the third seed in the Western Conference - and set a franchise record with 49 wins and 105 points.
The Stars had 107 points but were the sixth seed after finishing behind Anaheim and San Jose in the Pacific Division.
Notes - Defenceman Rory Fitzpatrick drew into the lineup after Brent Sopel was scratched with back spasms. ...The Canucks had six players on their roster who played their first NHL playoff game. . ...Prior to the game there was a ceremony honouring the 1982 Canuck team that went to the Stanley Cup final.