Islanders captain John Tavares had a goal and assist, and Nick Leddy had two assists in what could still be the franchise's final game at Nassau Veterans Memorial Coliseum. Cal Clutterbuck then got the loud crowd on its feet to seal the win with an empty-net goal with 53 seconds remaining.
John Carlson scored for the Capitals, who will host the deciding game Monday night. The series winner advances to face the Presidents' Trophy-winning New York Rangers, who finished off the Pittsburgh Penguins in five games Friday night.
Jaroslav Halak made 38 saves -- including 13 in the final period -- and bounced back from a 5-1 loss in Game 5 on Thursday night.
Kulemin's goal came during a frantic stretch in which officials allowed the play to continue despite players from both teams tangling in front of their benches.
Tavares started the play by driving into the Capitals' zone and attempting to split two defenders. He failed to get a shot off and then was bowled over heavily in the left corner by Alex Ovechkin and Karl Alzner.
No one seemed to notice that the puck squirted to Leddy at the left boards. He then fed Kulemin, who was all alone in front. Kulemin went across the top of the crease and wrapped the puck inside the left post.
Capitals forward Jay Beagle missed on a golden opportunity to tie it with 5:33 left.
Halak stopped Troy Brouwer's initial shot, and the rebound dribbled into the slot. Beagle got to it and lifted a shot over Halak but bounced it off the crossbar. The puck landed in the crease, where Islanders defenseman Johnny Boychuk shoveled it under Halak to stop the play.
Tavares opened the scoring 7 minutes in. After Islanders forward Ryan Strome was checked off the puck by Brooks Orpik, Tavares followed up and drove the zone up the right wing. With Carlson backing in, Tavares took advantage by cutting into the middle a snapping a shot in.
The Islanders were called for three penalties in the opening period, and the third one cost them. Carlson tied it at 1 with 5 seconds left and Tavares in the box for slashing Evgeny Kuznetsov.
Brouwer started the play by getting a lengthy pass at the Islanders blue line, and driving to the net. Jaroslav Halak stopped Brouwer, and then shoveled the puck to his right, and directly to Ovechkin in the circle. Ovechkin fed a pass across to Carlson, who took a few steps in and snapped it through a crowd and in off Halak.
The Islanders have a 3-4 record in Game 7s. The Capitals are 3-9, and 2-7 when hosting a seventh game.
Unless the Islanders advance, the Game 6 victory will be their final game at the arena. The franchise that won consecutive Stanley Cup championships from 1980-93 and produced such stars as Mike Bossy, Denis Potvin and Billy Smith, is moving to the Barclays Center in Brooklyn for the start of next season.
The Islanders are leaving Long Island after 43 seasons.
NHL Commissioner Gary Bettman on Friday said the move was long overdue because the arena was outdated, and placed the blame on local politicians for failing to approve several proposed plans for a new arena.
One fan carried a sign that read: "This Is Home, Not Brooklyn."
After the second period, fans inside the arena booed a commercial pitching season tickets for next season.
The Islanders and the NHL stepped up security inside and outside the arena, including having officers patrolling the surrounding parking lots on horseback.
Capitals fans complained they had their cars -- bearing District of Columbia and Virginia license plates -- vandalized last weekend. And even Islanders fans can't get along. A fight between two fans wearing Islanders jerseys erupted in the stands a few moments before Nicklas Backstrom scored in overtime of a 2-1 win in Game 4.
Orpik had beer spilled on him while celebrating Backstrom's winner Tuesday.
Ovechkin made a joke to CSN Washington-TV that he might consider taking a chair home as a souvenir.
It was a very chippy game that even had both captains -- Tavares and Ovechkin -- get tangled up and exchange shoves at center on their way to their respective benches for a line change in the first period.
Ovechkin, who is well known for his willingness to throw checks, lined up Islanders forward Cal Clutterbuck along the boards that sent both players flying in the second period.
The Islanders were minus another defenseman after Calvin de Haan was hurt during the third period of Game 5. Matt Donovan made his playoff debut stepping in for Griffin Reinhart, who struggled on Thursday. Rookie Scott Mayfield, who spent all this season in the minors, took over for de Haan. ... The Isles also sat C Anders Lee, who was replaced by Colin McDonald. ... Carlson scored just the second power-play goal of the series -- both by Washington. The teams entered the game having converted just one of 22 chances.