The 23-year-old Poulin, who has played well in a backup role to starter Evgeni Nabokov, was sharp throughout as the Islanders improved to 7-9-3.
"I felt really good from the beginning," said Poulin, who won for the second time in six games this season. "I was seeing the pucks well all night and was glad to see us play a strong game as a team."
Predators right wing Patric Hornqvist ruined Poulin's shutout bid with his fifth goal of the season with 5:30 remaining.
Nashville (8-8-2) was playing the sixth game of a seven-stop road trip. The Predators lost 5-0 at both Winnipeg and New Jersey before visiting the Islanders. They conclude their journey at Pittsburgh on Friday.
"We were slow to the puck all night and they did a better job winning the 1-on-1 battles," Predators coach Barry Trotz said. "We're going through a tough time. We have to tighten up and put together better efforts."
Tavares opened the scoring with 4:30 left in the first period with a power-play goal, his eighth of the season. Tavares was alone in the slot when he took a pass from Frans Nielsen from along the left boards as Nashville's David Legwand served an elbowing penalty.
Tavares settled the puck before firing it past Predators goaltender Marek Mazanec, who was making his first NHL start. The 22-year-old Czech netminder was recalled from Milwaukee of the AHL last week to back up Carter Hutton, already the backup to starter Pekka Rinne who is out with a lingering hip infection.
Bouchard scored his fourth of the season with 2:37 remaining in the first, also with an assist from Nielsen. Okposo increased the lead to 3-0 at 7:07 of the second, rifling a slap shot from the left circle past Mazanec for his sixth of the year.
The Islanders collectively lifted their game as they were missing high-scoring left wing Thomas Vanek for the second straight contest. New York acquired Vanek on Oct. 27 from Buffalo for popular winger Matt Moulson. Vanek suffered an upper body injury early in last Saturday's loss at Columbus.
"I like that we got our speed going and put together a solid team effort tonight," said Tavares, who assisted on Okposo's goal and leads the Islanders with 22 points. "It was important that we came home and played a strong game like this. Hopefully we can build on it."
New York beat Boston 3-1 in their previous home game on Nov. 2 -- when Vanek scored his first goal with the Islanders -- then lost at Washington, Carolina, Columbus and Montreal.
The Islanders have had little success against the Predators through the years. The victory was New York's first against Nashville in regulation since April 2002 and their first win at Nassau Coliseum over the Predators since a 2-1 triumph way back on Nov. 7, 2000.
Nashville had been 8-0-2 in their last 10 meetings with the Islanders, including a 3-2 home win on Oct. 12.
"We played a solid game start to finish," Islanders coach Jack Capuano said. "We were disciplined and our goaltender played very well. This was a total team effort."
The Islanders host Los Angeles on Thursday before welcoming the Detroit Red Wings to Long Island on Saturday to complete the homestand.