Tavares, who had two goals and two assists in a win over Tampa Bay two nights earlier, also assisted on the Islanders' other goal by Matt Moulson.
Evgeni Nabokov, who went through free agency after refusing to report to the Islanders last spring and playing part of the season in Russia, made 29 saves in his first start for the Islanders.
The Rangers' goals came from Brandon Prust and Marian Gaborik.
Henrik Lundqvist turned back 30 Islanders shots.
The Islanders have now won three straight. The Rangers were coming off losses to Los Angeles and Anaheim in Stockholm, Sweden.