The Islanders chose to go first in the shootout and after misses by Miroslav Satan and Viktor Kozlov, Robitaille beat Andrew Raycroft with a wrist shot.
Toronto missed all three of its chances against Rick DiPietro, who stopped shots by John Pohl, Mats Sundin and Alexei Ponikarovsky.
Trent Hunter scored twice in regulation for New York, while DiPietro stopped 27 shots. The win moved the Islanders two points ahead of the Maple Leafs in the race for an Eastern Conference playoff spot. New York has 68 points (30-23-8), but trails eighth-place Carolina by a point.