New York (16-22-7) kicked off a six-game road trip with their second win in as many nights after downing the Dallas Stars 7-3 on Monday.
It was another frustrating decision for Toronto (21-18-5), which outshot the Islanders 26-25 coming off a lopsided 7-1 home loss to the New York Rangers on Saturday night. That defeat snapped a three-game win streak.
The Leafs made it interesting when Nazem Kadri's shot deflected in off Joffrey Lupul at the 16:55 mark of the third period to cut New York's lead to 4-3. It was Lupul's 13th goal of the season, delighting the Air Canada Centre gathering of 19,164. But Clutterbuck cemented the win with an empty-net goal at 19:51.
Nielsen's power-play goal, his 16th of the season, broke a 2-2 tie at 3:41 with Lupul serving a goalie interference penalty. De Haan registered his first of the year at 6:01, his slapshot outside the Leafs' blue-line deflecting off captain Dion Phaneuf's sick and past goalie Jonathan Bernier, who wasn't sharp in his sixth straight start.
Bernier got the start despite allowing five goals on 32 shots against the Rangers.
Kyle Okposo and Michael Grabner also scored for the Islanders. Captain John Tavares, named to Canada's Olympic team earlier Tuesday, added three assists after registering five points (three goals, two assists) against Dallas.
Tyler Bozak and Mason Raymond had the other goals for Toronto, which fell to 14-10-1 at home.
Bozak opened the scoring, deflecting Paul Ranger's point shot past New York goalie Kevin Poulin at 5:05 of the first, his sixth of the season. Bozak has two goals and four assists in four games since returning from an oblique injury.
Poulin started ahead of regular Evgeni Nabokov, who was injured in Monday night's 7-3 win over Dallas. Nabokov's absence is a definite concern as the Islanders were 1-9-2 when he was out with a groin injury earlier this season.
Okposo pulled New York even at 19:16 with his 17th goal of the season, his wrist shot from well out eluding Bernier.
New York went ahead on a peculiar play early in the second. Bernier left his crease to stop a loose puck behind the Toronto goal but couldn't prevent it from going outside the no-play zone. That allowed Grabner to take the loose puck and score his sixth goal unassisted on the wraparound at 3:06.
Toronto had four power-play chances in the period but only scored at 8:25 to make it 2-2. Poulin stopped Jake Gardiner's point shot but Raymond converted the rebound for his 12th goal of the season but just his second in 16 contests.
NOTES — Defenceman Tim Gleason made his Toronto debut . . . Toronto beat the Islanders 5-2 in the first meeting of the season Nov. 19 at the ACC . . . The Leafs visit the Carolina Hurricanes on Thursday before heading to Washington on Friday night. Their next home game is Sunday against New Jersey . . . The Islanders placed Nabokov (lower body) on injured reserve after Monday's game and recalled Anders Nilsson from Bridgeport of the AHL . . . Okposo missed Monday's win over Dallas to be present for the birth of his daughter . . . Tavares extended his points streak to seven straight games but teammate Thomas Vanek's string ended at nine contests.