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 turned aside 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.
The Islanders have gained points in 13 of their last 15 games (9-2-4).
Ponikarovsky and Matt Stajan scored for Toronto, which is 7-2-3 since the all-star break. Raycroft was solid in the loss, stopping 28 shots.
Each team committed one major giveaway in the first period, but neither opportunity led to a goal. Toronto defenceman Bryan McCabe threw a cross-ice pass in front of his own net, where Mike Sillinger picked the puck off, but his shot hit Raycroft right in the chest.
DiPietro threw the puck directly at Alex Steen in the left circle and Steen quickly fired at the vacated net, but defenceman Chris Campoli covered just enough ice to get a piece of the shot and deflect it away.
The Islanders outshot Toronto 16-9 in the period, including seven shots during a power play.
DiPietro took his turn to rob Steen around seven minutes into the second period. Steen, all alone by the right post, sent a shot toward an empty net, but DiPietro dived and got his stick on the goal line and blocked the puck before it crossed.
Steen picked up an assist on the opening goal, scored by Stajan at 11:34. Stajan, skating straight up the slot, had Steen's pass deflect off his left skate between DiPietro's pads for a 1-0 lead.
Ponikarovsky made it 2-0 at 13:37 on a harmless-looking shot from the left boards. The puck deflected off the skate of defenceman Brendan Witt past DiPietro.
Hunter put the Islanders on the board with 57.1 seconds left in the period. He scored his 16th of the season when he tipped Sean Hill's right point shot past Raycroft.
Sundin had an apparent goal at 2:37 of the final period disallowed, as it was ruled there was incidental contact by one of the Maple Leafs with DiPietro in the crease.
Hunter tied it with 6:04 left in the final period on a tip-in of Tom Poti's left point shot, while New York was on its fifth power play of the night.
Raycroft kept the game tied with a brilliant diving pad save on Marc-Andre Bergeron with 4:50 left.
The Islanders had the best chance to score in the five-minute overtime, but Jason Blake's shot from the left side clanged off the right post halfway through the extra session.
Notes: New York has points in five straight home games against the Maple Leafs (4-0-1). ... Toronto entered with the most road points in the NHL with 38 (17-10-4). ... Islanders all-star forward Jason Blake welcomed his third child Wednesday as his wife Sara gave birth to their second daughter, Annabelle Marie.