With Edmonton holding a 3-2 lead, Calgary pulled goalie Miikka Kiprusoff and scored with 1.5 seconds left in regulation when Jay Bouwmeester's shot from the point was tipped in by Rene Bourque, who also had two assists.
In Edmonton's home opener on Saturday, goaltender Nikolai Khabibulin mishandled a puck with under a minute left to give the Flames a 4-3 victory.
"You can't lose two tougher games than that," Oilers captain Ethan Moreau said. "We played very hard in both games and did a lot of good things. I guess it gives some adversity to deal with early in the season because those are two of the toughest losses you are going to see all year, I hope."
Daymond Langkow and Dawes also scored in regulation for the Flames (4-0), who matched their best start since the 1993-94 season.
"That's a huge game for us," Dawes said. "Especially the way things ended up at the end there, scoring at the last second. We really battled hard right to the end. That's two big wins for us here. You never know what can happen, so that's whey you have to keep pushing.
"That's what we did tonight after falling behind by two."
Moreau, Lubomir Visnovsky and Ales Hemsky scored for the Oilers (1-1-1), who lost defenseman Sheldon Souray to a concussion in the second period. Patrick O'Sullivan had Edmonton's goal in the shootout.
Just 18 seconds into the second period, Souray was helped off the ice after going hard into the boards when he was tripped by Calgary's Jarome Iginila. Souray did not return to the game and the early diagnosis was a mild concussion.
Iginla said there was no intention to injure Souray.
"I wasn't going for the puck, I was just trying to hold him and ride him into the boards like they do 100 times," Iginla said. "I didn't mean to do that at all. I felt bad that he went in, but I went in pretty hard too and was a little dazed. It was freaky play and I definitely didn't mean to do that. He's an honest player and I didn't mean to hit him from behind."
Oilers coach Pat Quinn thought there should have been more of a response from his team.
"I don't understand the players of today," he said. "If that had happened in the old days he would have got hit over the head with a stick right after. It was a pretty dirty play in my opinion. He poked his feet out and then piled on top of him. Somehow they never deal with that ... and they won't let the vigilante stuff happen to deal with it. It's disappointing."
Calgary had a two-man advantage for 1:32 in the second period, but the Oilers did a great job on the kill, including a pad save by Nikolai Khabibulin on Dawes.
The Flames made it 2-1 with 29 seconds left in the second as Langkow sent a backhander toward the net that bounced off Jason Strudwick of the Oilers and deflected into the net. Calgary tied it 4 minutes into the third as Dawes tipped a shot by Staffan Kronwall past Khabibulin from the side of the net.
Edmonton regained the lead with 6 minutes left in regulation as Shawn Horcoff's shot from the slot was stopped by Kiprusoff but came out in front to Hemsky, who stuffed it into the net.
Moreau returned after missing one game with an infected ankle. D Steve Staios missed the game after getting his bell rung in Edmonton's shootout win over Dallas on Tuesday. Staios was replaced by Jason Strudwick. ... Calgary D Cory Sarich remained out with a foot injury. ... The Calgary power play came in with a 71.4 efficiency rating. The Flames were 0 for 4 with the man advantage Thursday. ... The Flames and Oilers meet for the third time this month in Calgary on Oct. 24.