Gaudreau shot short side on Jake Oettinger after he made an initial save on Elias Lindholm's chance in the slot.
"It came right down to the end there in Game 7 in overtime," said Gaudreau, the third player in Flames history to score in overtime of a Game 7. "You dream about stuff like that, scoring in a Game 7 in overtime.
"I was just trying to put it on net. I mean, throughout the whole series, most of the time you put it on net it's not going in against this kid. I got a good look, and it went over his shoulder there."
Oettinger made 64 saves, becoming the second goalie to make 60 or more saves in a Game 7 since 1955, when shots on goal were first tracked (Kelly Hrudey, 73, New York Islanders, 1987 Eastern Division Semifinals).
"He played a heck] of a game," Dallas captain Jamie Benn said. "Haven't seen too many like that. I mean, if it wasn't for him, we wouldn't even have been close to overtime or having a chance to win. He's a phenomenal young goalie. He's going to be great for this organization for a long time."
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Calgary will play the Edmonton Oilers in the second round. It will be the sixth postseason edition of the 'Battle of Alberta,' and the first in 31 years.
"I've been here for nine years and never had even a sniff of a chance to play them in the playoffs," Gaudreau said. "It's pretty special and it'll be a lot of fun. It's going to be good for the province. It's going to be fun for them, for us. It's going to be a pretty cool series."