The Oilers will face either the Colorado Avalanche or St. Louis Blues in the Western Conference Final. Edmonton had not advanced beyond the second round of the Stanley Cup Playoffs since 2006, when they lost to the Carolina Hurricanes in the Stanley Cup Final.
"We all want to win," McDavid said. "Everybody wants to win. I think our group has been through a lot this year and in years past. I think it's just made us hungrier and hungrier to be successful. We've had to learn a lot of lessons along the way, and it's got us up to this point, and obviously we want to keep going here."
Zach Hyman had a goal and two assists for the Oilers, the No. 2 seed in the Pacific Division. Mike Smith made 32 saves.
"We're only halfway, but it feels good to take that next step for now," Draisaitl said. "It feels good to see we grow as a team, we grow as an organization. I'm really proud of the guys. This was a hard series. That's a really good team over there and they made it hard on us and they pushed back, and we had to push back twice as hard. Great job by everyone."
Mikael Backlund had a goal and an assist for the Flames, the No. 1 seed in the Pacific. Markstrom made 30 saves.
"Right now, it's just all disappointment, frustration, upset," Backlund said. "We had a great team. A great group of guys. Proud of all the guys and would go to war with all the guys again. It just didn't finish the way we wanted."
RELATED: [Complete Flames vs. Oilers series coverage]
Andrew Mangiapane put Calgary up 1-0 at 10:13 of the first period with a one-timer from the slot for its first lead in 155:00 (Game 2), and Backlund made it 2-0 at 5:41 of the second period when he tipped Michael Stone's shot over Smith's left shoulder.
Darnell Nurse cut it to 2-1 with a one-timer from the point through traffic at 7:40, and Jesse Puljujarvi tied it 2-2 at 9:56 when he swatted in a rebound after Hyman's initial shot off the rush.
The Flames also blew a two-goal lead in a 5-3 loss in Game 2 after surrendering two four-goal leads in an eventual 9-6 win in Game 1.
"They just made plays in key moments of games, and we didn't," Calgary forward Blake Coleman said. "You look at a lot of the games, they were tight in the third and we just couldn't come up with that big play to put us over the hump in the last couple of games.
"It's such a small margin of error at this time of year, and that's part of the learning curve and the growth is learning how to win tight games and being a really difficult team in important situations."
Hyman gave Edmonton a 3-2 lead at 14:57 when he tapped the puck across the line at the side of the net, but Johnny Gaudreau tied it 15 seconds later at 15:12 after circling out from behind the net to make it 3-3.
Calle Jarnkrok put the Flames up 4-3 at 15:28, but Evan Bouchard tied it 4-4 at 16:08.