Tavares’ goal was a ping-pong shot that hit Francois Beauchemin and squeezed over the left shoulder of Jean-Sebastein Giguere and under the crossbar.
Less than two minutes before the winner, Kris Versteeg went in alone on a breakaway but the puck rolled on him and he shot it high and wide of the net.
Matt Moulson gave the Islanders a 1-0 lead at 2:17 of the second period before Toronto tied it with 2:05 left in the game on a Phil Kessel power-play goal.
New York [3-1-2] lost leading goal scorer Josh Bailey early in the first period with a suspected right leg injury.
Dwayne Roloson [2-1-0] was outstanding, stopping 29 shots for the win. In his last victory against Toronto, he stopped 58 shots in a 4-3 overtime win on Nov. 23 of 2009
Toronto [4-1-0] altered its line up, dressing Brett Lebda in favour of Carl Gunnarsson.
With his third period goal, Kessel leads the team in scoring with seven points.
Giguere [3-1-0] saw minimal action, stopping 16 shots in the loss.
No goals lots of action
The opening period was all about speed so it wasn’t surprising to see the Leafs top trio of Kessel, Tyler Bozak and Versteeg dominate for long stretches of time.
They generated a handful of half chances early on but kept at it and got the best opportunity of the period nine minutes in.
Bozak found Kessel alone in the low slot and fed him a backhand pass but Roloson beat the Leafs winger with a right pad save.
It was a free-flowing period with very few whistles. At one point, the play went 6:52 without a stoppage.
One goal in the second
The Islanders opened the scoring at 2:17 of the second period just seconds after a Toronto penalty expired.
Moulson took a cross slot pass from Tavares and forced Giguere to his knees before skating around him and depositing the puck in the top half of the net.
Versteeg had a golden opportunity to tie the game at 14:48 of the second period.
He had the puck at the top of the New York crease but couldn’t flip it over the right pad of a prone Roloson.
The hard luck Versteeg was denied his second goal of the season with less than four minutes left in the second.
Dion Phaneuf’s booming shot glanced off Roloson’s shoulder and hit the stick of Versteeg before he swept the puck inside the right post. The referee judged that Versteeg had his stick above the crossbar when the puck hit it, resulting in a no goal call.
Leafs tie it late
The chances came fast and furious early in the third period.
Just over two minutes in, Kessel took a stretch pass and roared down the right wing and blasted a shot into the stomach of Roloson.
Over two minutes later, Versteeg spotted Kessel hustling alone for the right post but his pass was in the skates of the Leafs winger.
The puck headed the other way and the Isles almost had their second goal of the night.
After Giguere made a save and slid behind the net, the puck was thrown out into the crease where Mike Komisarek deflected it toward the open net but he reached back and knocked the puck off the goal line.
The Leafs nearly tied it with less than 10 minutes left in the third but video review overturned the call on the ice.
Colby Armstrong knocked the puck behind Roloson where it hit the right post and slid across the line before Tavares knocked it out of the crease.
Roloson turned up large with 4:30 left in the game when he went left to right and robbed Mikhail Grabovsky with a right shoulder save.
Toronto tied the game with 2:05 left on the power play, setting up the overtime.
Phaneuf’s shot from the point bounced off the leg of Zenon Konopka and bounced to Kessel where he found the far side with a quick wrist shot for his fifth of the season.