The Oilers have now alternated wins and losses over their past nine games. They came into the game against Carolina having lost their prior game 2-1 to the Calgary Flames.
Eberle converted Taylor Hall's pass in front of the net to pull out the victory for Edmonton, which surrendered a three-goal lead to the Hurricanes.
"It's not the way we want to play. We can't be happy with that third period," Eberle said. "But we found a way to win and got the two points. It was a weird game. The crowd was with us and then against us and then back with us. Obviously, there were some plays in that game we would like to take back, but we won the game and got the two points. It was a relief to get the win."
The Oilers scored three first-period goals and added another early in the second to take control of the game before the Hurricanes stormed back with three unanswered goals to send the game to overtime.
"We played extremely well in the first and for a little bit in the second, and then completely took our foot off the gas," Eberle said. "I don't know what it was, but we just started turning pucks over and losing battles and playing bad defensively. It's frustrating when that happens, but we found a way to win and that's the biggest thing."
Hall opened the scoring 1:33 into the game, taking a pass from Sam Gagner and snapping a shot past Ward. It was Hall's 12th goal of the season and fifth in his past three games.
Ruutu tied the game on the power play at 6:19 with Nugent-Hopkins serving a hooking penalty. Ruutu found the puck in front of the Oilers net and slipped it past Dubnyk 14 seconds into the man advantage.
Yakupov restored the Oilers' lead at 11:04, taking a pass from Hall in front of the net and firing a shot past Ward. It was Yakupov's first even-strength goal of the season and fifth of the year.
Nugent-Hopkins put the Oilers up 3-1 at 18:14, left alone in front and converting Eberle's centering feed. The Oilers went on to outshoot the Hurricanes 14-6 in the first period.
"Obviously, it was not the start we wanted at all. We kind of played right into their hands," Eric Staal said. "They have skill and firepower, and we knew that. But it's not going to be perfect every night, and we showed a lot of resiliency coming back and getting it to 4-4 and having some good chances to win it in regulation despite all the penalties we had to kill. It's tough to lose it in overtime when you battle back to tie the game."
The Hurricanes were playing their fourth game in six nights and on the second half of back-to-back set. They lost 2-0 to the Vancouver Canucks on Monday.
"We had trouble finding our legs, and our execution wasn't at the level that it needed to be and they took advantage," Eric Staal said. "They're an up-tempo team, and they were good to start. We got to our game later and started feeling better with our jump and started building momentum. It's good to battle back to get a point, but it's unfortunate we had to kill a penalty in overtime."
The Hurricanes eventually did find their skating legs in the second period, but not before Joensuu put the Oilers up 4-1 at 1:40, benefiting from Mark Arcobello's shiftiness behind the Hurricanes net. Arcobello was able to elude a check then find Joensuu unattended in front of Ward.
Jordan Staal cut the lead to 4-2 at 9:37, finding a loose puck in front of the Oilers net and then room underneath Dubnyk's pad.
At 7:47 of the third period, Eric Staal scored a shorthanded goal to get the Hurricanes to within 4-3. The Hurricanes captain accepted a gift from Eberle, who was trying to play the puck back to the point on the power play. Staal raced in on Dubnyk, faked a shot, then stepped around the goaltender and fired it into the empty net.
"It got us to within one," Eric Staal said. "It got the bench going. It gave us some life. I felt like we were coming; the momentum was starting to turn our way. It was nice to get one for the guys and then we got the big one to tie it."
The comeback bid took a hit when Hurricanes center Riley Nash was assessed a five-minute major and a game misconduct for boarding at 9:19. Nash drove Corey Potter into the boards from behind, temporarily knocking the Oilers defenseman out of the game.
The Oilers were unable to capitalize on the power play and paid for it when Skinner tied the game at 14:46, firing a point shot that found its way through traffic and past Dubnyk.
Justin Faulk took a boarding penalty against Nugent-Hopkins early in overtime to give the Oilers the power play, which eventually led to the winning goal.
"I'd love to have a win every night," Oilers coach Dallas Eakins said. "I don't care how we get them. It was two points. Would we have like to have ended that game 4-1? Absolutely. But the important thing is there is two points in the bank."
The Hurricanes (13-13-6) continue their four-game road trip against the Calgary Flames on Thursday. The Oilers (11-18-3) host the Boston Bruins in the final of a five-game homestand Thursday.