The Oilers got off to a rough start with the Islanders scoring two quick goals in the first.
Nick Leddy hammered home a shot past Cam Talbot just 13 seconds into the game, assisted by Cal Clutterbuck and John Tavares.
Shane Prince scored at 5:49 to make it 2-0 for the Islanders, assisted by Dennis Seidenberg.
The Oilers needed to settled down after their scrambly start. Following a decent shift by their fourth line, Edmonton got their first of the game at 8:22.
Matt Benning fired a shot inside from the point and Patrick Maroon deftly tipped it in past Jaroslav Halak to make it 2-1. The helper was Benning's first NHL point.
"I thought the first five minutes, we weren't ready at the drop of the puck. After that, the last 15 minutes of the first period, I thought we got pucks deep, we cycled, we shot pucks, we got through and got second and third opportunities. We got rewarded after that," said Maroon.
A tripping penalty provided the Oilers with an opportunity to tie a few minutes later, and they made good on it.
Leon Draisaitl scored his first goal since October 14, on the man advantage, at 12:13 of the first. Benning recorded an assist to give him his first multi-point game of his young career.
The Oilers were shorthanded for two full periods, missing Jesse Puljujarvi and Zack Kassian. Both did not return with the team to the ice at the start of the middle frame. The team was already down one of their more consistent defencemen as Kris Russell missed the game with an undisclosed injury.
Down two skaters in this game, Edmonton went down a goal again to the Islanders at 11:18 of the second period. Casey Cizikas buried a rebound for the 3-2 Islanders lead.
"The depth we have in this lineup is the best I've seen," said Oilers winger Jordan Eberle. "It seems like every night (the fourth line) is scoring or coming up big for us. Tonight the power play was great. We had two tonight. It's different stories for different wins every night. We're finding different ways to win."
Milan Lucic tied the game again for the Oilers with his fifth goal of the season, on the power play, at 16:50 of the second. Eberle had the shot, and Lucic buried the rebound.
"We're a mentally stronger team here," said Eberle. "We had a little more pushback and, obviously, tied the game. Then we were down 3-2 and decided to come back and win in overtime. That's a sign of a more mature team."
The game went to overtime following a scoreless third. In the overtime period, Edmonton had a few great chances.
39 seconds into the overtime period, Halak made a save on Eberle, who was alone on the doorstep. Then Connor McDavid could not finish an Eberle feed, after being denied at the post. There were nine combined shots in the five-minute overtime.
"I'm sure it was exciting," said Eberle. "That's why they put the format in. There's bound to be odd-man rushes and turnovers and breakaways. That's what (the fans) got tonight. You've got to tip your hat to their goaltender and Cam too. They made some big saves in overtime."
Letestu was the lone shooter to score in the shootout, burying a shot glove side of Halak.
Edmonton (8-3-1) plays Detroit Sunday on the second half of a back-to-back.