His third-period goal stood up as the game-winner in the Oilers' back-and-forth 6-3 victory against the Carolina Hurricanes at Rexall Place.
Draisaitl skated the length of the ice and jumped into the arms of Oilers defenseman Andrew Ference after scoring, slamming the captain onto the ice.
"It was really exciting. That's something I think every hockey player as a kid dreams of," Draisaitl said. "To make this happen tonight with this team is even more special for sure. It was an unbelievable play by David [Perron].
"The two guys that I have on my side (Perron and Teddy Purcell) are the two most-skilled guys I've ever played with. I feel like sometimes they're a little too skilled for me. It was a great play. The first shot, [goalie Anton Khudobin] kind of robbed me, and I kind of stayed with it and batted at the rebound. I have to work on my [celebrations], though, for sure."
Draisaitl, who turns 19 on Monday, scored at 12:55 of the third period after the Hurricanes had battled back from a one-goal deficit on three occasions.
Edmonton (3-4-1) has won three games in a row for the first time since Jan. 26-29. Carolina is 0-5-2 to start the season.
"We found a way to win in the third, and that's the most important thing," Eberle said. "We got the win, we got the two points. That's something we weren't doing last year. We have a little bit of a streak going here.
"It's definitely nice to get the monkey off my back. It seems like it's the same every year; you get one and I feel good and I start playing my game again. It's funny how it works; you get one and the net turns into a soccer net."
Goaltender Ben Scrivens made 27 saves to earn his third win of the season. Khudobin stopped 23 shots for the Hurricanes.
"We're a bit emotional in here right now," Harrison said. "We showed some resiliency in this game. We have to continue to move forward and work to put together a 60-minute game. No one is going to give us a game. We have to take a game from someone. Until we do that, we're not going to get it. We have to continue to build and get better."
Eberle scored his first goal to give the Oilers a 1-0 lead at 9:44 of the second period. He knocked down Harrison's clearing pass, gave the puck to Taylor Hall and headed to the net. Hall was able to lift a pass over to Eberle, who snapped it past Khudobin.
Nash, a first-round pick (No. 27) of the Oilers in 2007, tied the game at 15:08, redirecting a point shot from Justin Faulk past Scrivens.
Joensuu restored the Oilers lead at 16:46, knocking a rebound past Khudobin after the Hurricanes goaltender had stopped a shot from defenseman Nikita Nikitin.
Skinner scored his first goal of the season at 17:58 to tie the game 2-2. He gained the blue line, cut to the middle of the ice and took a shot that beat Scrivens, who appeared to be screened by Oilers defenseman Mark Fayne.
"We showed some character coming back as many times as we did," Skinner said. "We just couldn't find that one goal to put us up. It was tough the way it ended; it's tough to swallow."
Eberle scored his second goal at 4:26 to put Edmonton ahead by a goal for a third time. He knocked the puck off Hurricanes defenseman Brett Bellemore, picked it up at the right faceoff circle and snapped a shot past Khudobin.
Harrison tied the game 3-3 at 7:50 on a long shot from the point following a scramble in front of Scrivens.
Draisaitl put the Oilers up for good a little more than five minutes later. He took a cross-ice pass from Perron then lifted a backhand over Khudobin after the Carolina goalie had stopped his initial shot.
Nugent-Hopkins scored to make it 5-3 at 15:50 of the third off a scramble in front. It was Nugent-Hopkins' third goal in three games.
"I thought we did a pretty good job of weathering the storm," Nugent-Hopkins said. "… They kept tying it up. We just kept our noses to the grindstone, kept battling, and it didn't seem like there was any let-up once again."
Hendricks scored an empty-net goal at 18:33 after Hurricanes coach Bill Peters had pulled Khudobin in favor of an extra attacker.
"The effort was fine," Peters said. "We competed hard and did good things. It was a much better effort than [Thursday] night when we came unglued a little bit when we got behind in Calgary (a 5-0 loss). We showed a lot more composure and had some good things we can build off of."
The Oilers will play the fifth of their seven-game homestand Monday against the Montreal Canadiens. Carolina concludes a five-game road trip Tuesday against the Vancouver Canucks.