The Senators (12-7-4) have lost three in a row. They also have dropped eight of nine on the road.
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.
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. Defenceman Patrick Wiercioch's first career goal caromed off the stick blade of Islanders defenceman 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.
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