The Islanders trailed 3-1 in the second period before reeling off the next four goals en route to their seventh win in nine games.
Thomas Hickey and Colin McDonald scored 3 minutes apart late in the second to forge a 3-3 tie.
With 5:35 left in the third, the Rangers were whistled for having too many men on the ice, and Islanders went on a power play.
Vanek and Nielsen each finished with a goal and an assist. Tavares registered three assists for the Isles.
Matt Martin also scored and Kevin Poulin recorded 30 saves for the victory.
Rick Nash scored twice for the Rangers, whose three-game winning streak was snapped.
Chris Kreider added a goal with two assists and Talbot ended with 32 stops.