Kulemin made it 3-0 at 11:24. Nilsson was able to deny Frans Nielsen's wrist shot from the slot, but Kulemin, who went 12 straight games without a goal, followed the play and converted on the rebound.
Oilers coach Todd McLellan pulled Nilsson after Kulemin's goal in favor of Talbot. New York outshot Edmonton 16-7 in the first period.
"I'm really disappointed in the preparation, the approach to the game," McLellan said. "We got our butts handed to us in Montreal (a 5-1 loss on Saturday) and we came in here and responded in a very inappropriate way and that's very disappointing."
The momentum carried over into the second period when Okposo scored a power-play goal at 2:32. Moments after Greiss robbed Benoit Pouliot's breakaway chance, Okposo took a pass from Leddy and beat Talbot with a wrist shot from the right circle to make it 4-0.
Okposo scored again 3:37 later when his wrist shot from inside the blue line found its way through a crowd and past Talbot.
Greiss, who won for the 14th time in 22 games this season, raised his save percentage to .929.
"I just know that when he gets his opportunity, he takes the ball and he runs with it," Islanders coach Jack Capuano said of Greiss. "He's worked extremely hard. I just had that feeling he'd be there tonight, and he played well."
The Oilers finally solved Greiss when McDavid scored at 6:50 of the second period. Greiss made an initial save, but the rebound went off Islanders center Casey Cizikas to McDavid, who tapped it over the goal line to make it 5-1.