It was the sixth road win in a row for the Islanders, who forced the overtime when Brock Nelson's goal early in the third period evened the score.
Brian Strait helped set up the winner with a pass to Grabner in the Avalanche zone. Grabner skated across the slot before circling back and firing from between the circles to get the puck past Semyon Varlamov.
Kevin Poulin, who had 30 saves, won his fourth straight decision and improved to 5-1 in his last six games.
I think everyone left it out there. Every game this season, regular season and playoffs, we played as hard as we could. We're not out of the playoffs right now because of lack of effort, and that's all that we can ask of each other.
— Predators defenseman Seth Jones after Nashville was eliminated by Chicago in Game 6 on Saturday