The Islanders (23-16-5) came into the contest seventh in the Eastern Conference but moved to within two points of fifth-place Toronto (24-15-5) with the victory. New York overcame an early 2-0 deficit to win the first of five straight road games it will play to end the regular season.
But the Islanders have excelled on the road, improving to 13-5-2 away from home. They earned their fifth win in six games as they chase their first Stanley Cup playoffs berth since 2007.
The 22-year-old Tavares, a native of nearby Mississauga, Ont., who played his junior hockey with the Oshawa Generals and London Knights, sent a season-high Air Canada Centre gathering of 19,676 home disappointed while continuing to strengthen his case for consideration as a Hart Trophy candidate.
Cody Franson, Joffrey Lupul and Dion Phaneuf scored for the Leafs, who were solidly outshot 38-21, the fourth straight game and 10th time in 13 contests that has happened. Phil Kessel, appearing in his 500th career NHL game, added three assists.
Toronto will have to wait to clinch its first playoff berth since 2004. The Leafs could have ended their post-season drought with a win over the Islanders and a Carolina victory over the Winnipeg Jets, but instead dropped to 13-8-2 at the ACC with a second straight loss.
The Islanders controlled an eventful first, outshooting Toronto 19-5 but needed a late surge to erase a 2-0 deficit. Moulson's power-play goal, his 15th, at 15:03 tied it before Tavares, the first overall pick in 2009, gave New York its 3-2 lead at 17:24. Tavares fired a shot on a 2-on-1 break that Leafs goalie James Reimer got a piece of but couldn't keep out for his 25th goal of the season.
Boyes had earlier converted Tavares' smart centring pass for his ninth goal at 9:07 that cut Toronto's lead to 2-1.
Lupul opened the scoring at 5:05, skating around Islanders defenceman Matt Carkner and putting a shot through goalie Evgeni Nabakov's legs for his ninth of the season. It came after Toronto, which boasts the NHL's third-ranked penalty-kill unit, successfully defended a two-man New York power play for 1:28 early in the period, with Reimer making a nice save off Moulson.
Franson's power-play goal at 6:41, his fourth this season, gave the Leafs a 2-0 advantage as they scored on their first two shots of the period.
It was more of the same in the second as the Islanders held a 15-5 edge in shots and had the period's lone goal. Tavares scored his second of the game and 26th overall at 14:40.
Nielsen made it 5-2 with his fifth goal at 10:22 of the third before Phaneuf rounded out the scoring late on the power play.
Notes: Toronto and Buffalo are tied for the NHL's youngest roster with an average age of 26.9 years of age. ... The Islanders came in sporting a 20-2-2 record when they score at least three goals but just 2-14-3 when scoring two or fewer times. ... New York defenceman Andrew MacDonald entered Thursday's action with a league-high 114 blocked shots.