Washington trailed 2-0 early in the second period, then 3-2 after Richard Park scored for New York 2:12 into the third by splitting two Capitals players and flipping a backhand past Jose Theodore.
But that's when Green took over, raising his goal total to a career-best 30.
First, he took a cross-ice pass from Alexander Semin and sent a one-timer past Joey MacDonald 10:55 into the third period to tie it. Then, at 12:18, Green was barely inside the blue line when he flicked the puck past MacDonald.
Nicklas Backstrom added a final-minute goal for Washington, which completed its first season sweep of the Islanders since 2002-03.
The Capitals would clinch first place if the Carolina Hurricanes lose to the New York Rangers on Thursday. A year ago, Washington didn't seal its first division title since 2000-01 until its final game.
The Capitals are up to 47 wins, a total they haven't eclipsed since 1983-84, and have 101 points, a mark they haven't topped since getting 102 in 1999-00.
Kyle Okposo and Frans Nielsen each had a goal and an assist for New York. Okposo put the Islanders in front with his team-high 18th goal about 14 minutes into the game - and 12 seconds into a power play - off a pass from Nielsen.
New York went up 2-0 only 37 seconds into the second period when Okposo and Jeff Tambellini set up Nielsen's goal. The puck appeared to go in off Nielsen's skate, but the goal was allowed after a video review.
Then things got rough - and the Capitals began playing much better.
First, Washington's John Erskine and New York's Joel Rechlicz fought. About 1? minutes later, Green leveled Andy Hilbert along the boards, prompting a brawl involving nearly everyone on the ice.
Green wound up sitting alongside Erskine, whistled for charging, while New York's Tim Jackman and Washington's Jeff Schultz drew fighting penalties - as Capitals coach Bruce Boudreau yelled colorfully at the officials.
Something in that whole sequence got Washington going, and Ovechkin made it 2-1 at 5:37 of the second period. Sergei Fedorov went behind the net one way and suddenly sent the puck back in the opposite direction to a wide-open Ovechkin, whose shot bounced off MacDonald's left pad and in.
At 9:54, Keith Aucoin's first NHL goal since March 2008 tied it.
Notes: Islanders D Mark Streit, the team's leading scorer, sat out with a strained groin. He's day to day. ... Washington's Donald Brashear missed his ninth consecutive game with a knee injury, but the team expects the enforcer to be available for the playoffs. ... Capitals D Milan Jurcina was a healthy scratch. ... The Capitals signed F Stefan Della Rovere to a three-year, entry-level contract beginning next season. He was a seventh-round pick in the 2008 draft.
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