Paul Kariya ended the shootout with a goal after the Islanders chose to shoot first and had misses by Miroslav Satan and Viktor Kozlov.
Mike Sillinger scored New York's only goal in regulation, while Mike Dunham stopped 32 shots. Tomas Vokoun made 26 saves for Nashville.
Dunham has made three consecutive starts while usual starter Rick DiPietro recovers from a groin injury.
Nashville played its first overtime game of the season, while the Islanders were involved in their second. New York beat Anaheim in a shootout Wednesday night in California.
The Islanders wasted little time opening the scoring. Just as their first power play expired, Sillinger, down in the low right slot, took Mike York's pass-out and fired an uncontested backhand past Vokoun on the short side. The goal, Sillinger's second of the season and 200th of his NHL career, came at 4:38.
Dunham made a big save in the period when he snared Scott Hartnell's rising one-timer with his glove with 3:39 left.
Sullivan tied it just after the Predators' first power play expired at 10:15 of the second period. His one-timer from the left circle got past Dunham.
New York got a scare with 3:33 left in regulation when Dunham went down after taking Kimmo Timonen's slap shot off his mask. The goalie shook it off and stayed in the game.
Vokoun made his best save in overtime when he robbed Shawn Bates with his glove at the right post with 28 seconds left.
Notes: Islanders forward Alexei Yashin will play in his 800th NHL game when the Islanders host Pittsburgh on Thursday night. Nashville is one of only two teams to be scheduled for a three-game road trip against all the New York area teams in one sweep. Florida is the other. ... The announced attendance was 9,431.