"We battled hard and we were fortunate to win the skills competition in the shootout," Islanders coach Jack Capuano said.
The Senators (12-7-4) have lost three in a row. They also have dropped eight of nine on the road to fall to 3-6-2 away from home.
Nielsen scored to open the shootout against backup Robin Lehner. Tavares' goal came after Nabokov made his two stops.
The Islanders (9-11-2) improved to 3-8-1 at home after losing the first three in a seven-game homestand.
"I wasn't 100 percent but I gave it my best," said Tavares, who was described as "under the weather" by Capuano before the game. "I'm glad we got the win."
Tavares added that he had trouble sleeping the night before the afternoon game.
"John did what elite players do," Capuano said. "He found a way for us."
Michael Grabner's unassisted goal late in the third period put the Islanders up 2-1. Grabner intercepted a clearing attempted by the Senators and rifled the puck past Lehner for this eighth goal of the season at 16:31.
But Kyle Okposo was sent off for tripping with just under two minutes left, and the Senators capitalized on the power-play opportunity. Defenseman Patrick Wiercioch's first career goal caromed off the stick blade of Islanders defenseman Travis Hamonic and eluded Nabokov with 1:09 to go.
Both teams had excellent chances in overtime. The best save was a glove stop by Lehner on Hamonic from the right circle. Nabokov made a terrific save on Colin Greening in the opening shift of the extra five-minute session.
"It's disappointing but we did play well in the second and third periods," Senators coach Paul MacLean said. "Lehner did a good job for us and our penalty killing was very good."
Defenseman Mark Streit helped set up Boyes' goal at 7:48 when he rushed the puck up the ice and passed to left wing Matt Moulson, who sent it back to Boyes skating through the slot. He sent a wrister over the glove and left shoulder of Lehner for his fifth of the season.
The Senators played the third-stringer Lehner for the second time this season. Starting goaltender Craig Anderson is out with a sprained right ankle and backup Ben Bishop played Saturday in a 2-1 loss at Philadelphia.
Zibanejad tied it at 8:36 of the second. Nabokov gingerly shot the puck around the right boards and it was intercepted by Chris Neil, who shot it back toward the net. Hamonic then tried to clear the puck away but instead put it on the stick of Zibanejad, who shot it past Nabokov for this fourth of the season.
The 19-year-old Zibanejad was Ottawa's first choice and the sixth overall in the 2011 draft.
The Islanders had lost four straight and six of their past nine against the Senators, including a 3-1 loss in Ottawa on Feb. 19 when goaltender Rick DiPietro made his final start before being waived and sent to AHL Bridgeport.
Capuano emphasized that despite the problems at home, he remained confident that his team's work ethic would generate positive results.
"We have nine wins but we can do better," Capuano said. "This game is over and now we're focus on Montreal for Tuesday. We have to try to be better every day."
Ottawa was without D Mike Lundin, who suffered a concussion Saturday on a hit by Philadelphia's Harry Zolnierczyk ... Alfredsson played his 1,153rd game for the Senators. He is the longest serving captain in the league (appointed in October 1999) and is the only player to record 1,000 points with the franchise.