The first period went by without a goal, thanks to the two netminders between the pipes. Thomas Greiss wiped the Oilers first power play of the game off the board, stopping Mark Letestu's shot from in front.
At the Oilers end, Cam Talbot was locked in during the opening frame. He made an early glove save on Josh Bailey, who was riding a nine-game point streak, less than a minute into the game. He then robbed Anders Lee from point-blank range on an Islanders power play late in the period.
Talbot finished the game with 36 saves on 37 shots.
"This is a tough building to come play in," said Talbot. "They've played extremely well at home. This should give us some confidence moving forward. We started this trip off right. We have to continue that moving forward. We have a long ways to go."
With less than two minutes to play in the first, Talbot squared up to face a Ryan Pulock slap shot and he squeezed the pads to keep things scoreless.
Edmonton had a big opportunity to take the lead at 18:44 of the first, when Casey Cizikas went to the sin bin for tripping McDavid. The Oilers, however, were unable to capitalize.
They did break through first on the scoreboard, taking a 1-0 lead at 1:59 of the second period. Moments after the Islanders hit the post, Leon Draisaitl found himself on a breakaway. The forward snapped a shot on net that flipped past the pad of Greiss. Milan Lucic recorded the helper.
"I thought we played a very solid game," said Draisaitl. "There's still things we need to clean up, but right now it's about getting points and getting our game back in the right direction. Tonight was a very good step."
The Islanders evened the score at 4:43 of the middle frame. Former Oiler Jordan Eberle took a drop-pass and shot the puck from the faceoff dot high on Talbot, burying his fifth of the season.
For the seventh time this season, the Oilers went into the third period with the score tied after two.
The third period started in favour of the Islanders, with the home team controlling the play for nearly the whole first half of the frame.
With Oilers winger Drake Caggiula unavailable in the third, Zack Kassian stepped into his spot on the Ryan Nugent-Hopkins and Milan Lucic line. The trio generated the team's best chances of the first half of the third.
Talbot shut the door in the third, stopping all 13 Islanders shots he faced in the frame. The Oilers had just two shots of their own in the third.
"He was great," said McDavid. "I didn't realize it was that lopsided in shots but they had some chances for sure in the third and he did a great job."
In overtime, McDavid got loose on a 2-on-1 with Draisaitl trailing. McDavid dropped it back and then Draisaitl fed the captain with a backhanded pass at the side of the net and McDavid buried the one-timer for his sixth goal of the season.
"I had a little bit of a break," said McDavid. "That Leddy is a pretty incredible skater and he was able to get in front of me so I dropped it for Drat and realized we had a 2-on-1. He made a great play back."
The Oilers road trip continues Thursday in Newark, as the team takes on Taylor Hall and the New Jersey Devils.