The Hurricanes had a good opportunity to tie it with 3:45 remaining, but Talbot made a save against Victor Rask on a breakaway following a turnover by Oilers defenseman Adam Larsson.
Edmonton then killed a high-sticking penalty against Larsson at 16:24 of the third.
Slepyshev opened the scoring at 1:51 of the first period when his wrist shot from the top of the left faceoff circle beat goalie Cam Ward, who had 25 saves.
Eberle made it 2-0 at 9:22 of the first when he banked in a shot off Carolina defenseman Ron Hainsey, and Pitlick scored at 2:35 of the second to give Edmonton a 3-0 lead.
"When you go down early on the road, you want to show some composure," Skinner said. "We had a little miscommunication on some of the goals and we just shook those things out and sort of learned from them. On the road, it's tough to get down. We did a good job battling back and we put ourselves in a good situation with a power play with three minutes left. You'd like to capitalize and tie the game there."
Stempniak scored at 3:18 of the third period to cut the Oilers lead to 3-1, putting a rebound past Talbot. He scored again at 4:14 to pull the Hurricanes to within at goal.
Oilers coach Todd McLellan challenged the goal, claiming the Hurricanes were offside, but it was upheld by video review.
"It wasn't pretty, to get up three and give it back to them was not part of the game plan," McLellan said. "Neither was giving up a number of free looks at our net in the first seven minutes of the game, but we had a good bounce-back night from Cam. If he wasn't the First Star, he should have been. We're really happy for him."