Tkachuk shot five-hole on a partial breakaway to extend the lead to 8-6 at 8:55.
"We played probably the worst 15 or 20 minutes of hockey we have all year, definitely in the playoffs, and it's still a tie game," Tkachuk said. "Wasn't the end of the world. Still got to go out there. Can't roll over and let them win. It was big to get that seventh goal from 'Ras,' and we went from there."
Tkachuk scored into an empty net at 17:49 for the 9-6 final.
"We got to 6-6, then turned over some pucks and it ended up in the back of our net," Woodcroft said. "Like I keep saying, we scored six goals on the Calgary Flames in their building, that should be enough to win a game."
Elias Lindholm gave the Flames a 1-0 lead 26 seconds into the first period when his shot went in off the glove of Smith, and Mangiapane scored 25 seconds later off a pass by Mikael Backlund from behind the net to make it 2-0.
The two goals scored in 51 seconds are the fastest to start a playoff game in NHL history.
"When you're down 2-0 before for it hits the 19-minute mark on the clock, it's obviously not a good start," Draisaitl said. "Clearly, we weren't ready."
RELATED: [Complete Flames vs. Oilers series coverage]
Brett Ritchie extended the lead to 3-0 at 6:05 after stealing the puck from Evander Kane in the Oilers zone.
McDavid cut it to 3-1 at 7:41, stickhandling at the top of the crease and shooting five-hole after the puck popped out to him from below the goal line.
Coleman scored on a rebound 45 seconds into the second period to make it 4-1 before pushing the lead to 5-1 at 6:10 when he redirected in a slap pass from Noah Hanifin in the slot.
"There's a lot of different ways to contribute in the playoffs, and I try to find different ways to do it when I'm not scoring," Coleman said. "Obviously, on a night like tonight, it's a good thing pucks were going in, because we needed every goal we could get."
Evan Bouchard made it 5-2 at 7:10, but Tkachuk batted the puck past Koskinen on the power play at 8:24 to make it 6-2.
Hyman scored from the outer edge of the left circle to make it 6-3 at 9:38. He then cut it to 6-4 at 14:06 when he skated out from behind the net and shot glove side on Markstrom.