The Oilers had led 4-1 at one point, but allowed the Hurricanes back into the game.
The Oilers' forward was able to remain on the ice and set up the winner, sending a pass to Eberle at the side of the net to put the game away at 1:48 of OT.
"I'd love to have a win every night. I don't care how we get them," Eakins said. "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."
Eberle said it was a good feeling to come away with the victory, especially after almost letting it slip away.
"It was a relief to get the win," he said. "We played extremely well in the first and for a little bit in the second and then completely took our foot off the gas. 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."
Taylor Hall, Nail Yakupov, Nugent-Hopkins and Jesse Joensuu also scored for the Oilers, who have gone 7-3-1 in their past 11 games. Edmonton also has a 2-1-1 record through the first four games of a five-game homestand.
"We are working hard to get out of the hole we created for ourselves and it seems we are on a bit of a run now," Hall said. "Now it is time to get two in a row, something we haven't been able to do for a while."
"We're a pretty resilient team," Carolina coach Kirk Muller said. "We have to look at the positives, we got a point. We got outplayed in the first, and came back in the second. We had a really good second and third. We got a really big point and possibly could have had two."
The Oilers got on the board just 1:33 into the first period.
Hall had a clear lane to the net and seemed to surprise Carolina goalie Cam Ward with a quick shot that beat him glove-side. It was Hall's 12th goal of the season and fifth in the past three games.
The Hurricanes tied the game just over six minutes into the opening period on a power-play goal. Ruutu was able to tip a Skinner shot on net and then get his own rebound after Edmonton defender Nick Schultz failed to clear the puck in front and put it past Oilers starter Devan Dubnyk.
Edmonton vaulted back in front with nine minutes remaining in the first period as Yakupov took a pass from Hall and sent a blast from the slot that went through the legs of both defenseman Ron Hainsey and Ward. It was just the fifth goal of the season for the 2012 No. 1 draft pick.
Yakupov was looking for another with three minutes left with another shot from in tight, but the puck went off the post.
Edmonton kept coming and made it 3-1 with 1:46 remaining in the first period as Nugent-Hopkins was able to beat Ward with a shot up high while sliding in toward the net on his knees.
Edmonton outshot the Hurricanes 14-6 in the opening 20 minutes.
The Oilers added to their lead less than two minutes into the second period as Mark Arcobello worked hard to get a backhand away off his knees from behind the net to set up Joensuu for a quick shot on the doorstep and a 4-1 Edmonton advantage.
Carolina started to push its way back into the game and was rewarded midway through the second. After sustained pressure in the Edmonton zone, a puck came loose to Jordan Staal in front of the net and he scored his seventh of the season through Dubnyk's legs.
The Hurricanes made it a 4-3 game eight minutes into the third period, as Eric Staal was able to pick off a poor Eberle pass and streak down the ice on a short-handed breakaway. Staal faked a slap shot and got Dubnyk to commit and go down before calmly putting the puck into the empty net behind him.
"We definitely lost some momentum there and they gained some," Nugent-Hopkins said. "We can't let that happen. When you get a five-minute power play, you have to get at least one goal. That wasn't our best job. But it is a big win for us and we have to build off this now."
The lack of offense proved costly as the Hurricanes tied the game with just over five minutes remaining in the third after Edmonton failed to clear the zone in a sloppy sequence. Skinner was able to take a no-look turnaround shot from the point that got past Dubnyk for his 10th goal of the season to make it 4-4.
Both teams return to the ice on Thursday as the Hurricanes play the third game of a four-game trip in Calgary against the Flames, while the Oilers wrap up a five-game homestand against the Boston Bruins.
It was the first of two meetings this season between the Oilers and Hurricanes, who haven't played a game against each other since December 2011. It was only the 19th regular season meeting between the two teams, who met in the 2006 Stanley Cup final. Their next game will be on March 16 in Carolina. ... The game featured the NHL's top two stars from the past week. Skinner was the league's first star after scoring five goals and adding an assist. Hall was the second star with four goals and two assists.