Edmonton went 2-for-5 on the power play.
"We took too many penalties," Islanders coach Barry Trotz said. "It's not a winning combination when the only two things that you did for the most part tonight was win face-offs and turn pucks over."
The Oilers chased Varlamov when Zack Kassian made it 4-1 at 8:46 of the second.
Neal scored his fourth of the game from the bottom of the right circle to make it 5-1 at 6:51 of the third period.
"[Neal] found his groove there," Tippett said. "He had 11 or 12 attempts at net, so good for him, get on the board. I give him credit; he's come in here, he's a real energized player. He's helped our group, not just scoring some goals, but he's really helped our group with giving some juice in the locker room. Happy to see him doing well."
Martin scored at 13:32 of the third for the 5-2 final. Nelson had a goal taken away at 16:48 when video review determined the puck was kicked in.
"We certainly didn't quit, so that's a positive," Martin said. "But we've got get out of the gates better. We haven't really come out of the gates in the first period and played well at all yet. In our building, that's usually our forte. We've got to do a better job of that."