Kuznetsov, who five months ago had trouble holding down a place in the Capitals team, proved to be the secret weapon against the Islanders, notching a team-high three goals across the seven games of the series.
After the teams traded early goals in a tense game, Kuznetsov weaved through two Islanders defenders before ripping a wrist shot over New York goaltender Jaroslav Halak with 7:18 remaining in the third period.
The Capitals advance to face the Rangers in the playoffs for a third time in four years and will be seeking revenge after they eliminated Washington in seven games in 2012 and 2013.
Right winger Joel Ward also scored for the Capitals, while center Frans Nielsen netted the lone goal for New York.
Tyler Johnson scored twice as the Tampa Bay Lightning forced Game Seven of their Eastern Conference first round series with a 5-2 victory over the Detroit Red Wings at Joe Louis Arena.
Center Alex Killorn, defenseman Jason Garrison and center Cedric Paquette also scored for Tampa Bay. Right winger Nikita Kucherov had three assists and goaltender Ben Bishop made 22 saves.
Left winger Tomas Tatar had two goals for Detroit while goalie Petr Mrazek stopped 23 shots.
Tatar brought the Red Wings to within 3-2 early in the third period with his second goal of the game but Killorn scored with a little more than five minutes remaining to restore the two-goal cushion. It was his second goal of the playoffs.
Paquette added an empty-netter in the final minute to complete the scoring.
Game Seven is in Tampa Bay on Wednesday.