But the Washington Capitals were defeated by the Ottawa Senators 4-3 in overtime, which means the Islanders will have to wait at least one more day before seeing an "X" next to their name in the standings.
"It's amazing the races that are going on in the East and West," Islanders coach Jack Capuano said. "You can't control that. You can control what you can control and that's all we are focused on right now."
New York (46-27-6) moved into second place in the Metropolitan Division, one point ahead of Washington. The Pittsburgh Penguins were defeated by the Columbus Blue Jackets, 5-3; two points behind the Capitals, the Penguins hold the first wild card into the playoffs from the Eastern Conference.
Halak made 21 saves for his 37th win and sixth shutout of the season. He is 7-2-2 with two shutouts against the Sabres in his career.
"If you would have told us that at this point with 98 points, we'd still be battling [for a playoff spot], it's very surprising," Capuano said.
The Islanders don't play until Tuesday against the Philadelphia Flyers but can clinch Sunday if the Senators, who visit the Toronto Maple Leafs, lose in regulation. New York also controls its destiny; if the Islanders win their remaining three games, they will have home-ice advantage in the Eastern Conference First Round.
"It would be nice to get home-ice advantage and let our fans have more opportunity to watch us here," defenseman Travis Harmonic said.
Okposo scored a power-play goal at 6:37 of the first period.
New York defenseman Brian Strait took a double-minor for high-sticking at 16:33. Halak made two saves during the power play, including stopping a Zach Bogosian slap shot from the left circle with 1:39 left.
"I think guys did a good job killing the penalty," Halak said. "Everyone was competing, blocking shots, being in the right spots. Obviously it was a big kill for us at the end of the second."
New York has allowed one power-play goal on 34 chances in its past 15 games.
Nelson scored at 9:32 of the third period to give the Islanders. Bogosian tried to clear the puck and it ricocheted off a stick that was on the ice and went to Nelson near the left circle, where he sent a wrist shot past Sabres goalie Andrey Makarov for his 20th goal.
"Just unfortunate," Bogosian said. "Off the stick and right to the guy. It happened pretty quick."
Kulemin, playing his 500th NHL game, scored his 14th goal off a rebound of a Hamonic shot from the right side at 13:13 of the third period.
Buffalo (22-49-8) lost to the Chicago Blackhawks 4-3 on Friday.
"No matter where they are in the standings … they compete hard," Islanders captain John Tavares said. "We knew this was a game we wanted to have."
Makarov made 33 saves in his NHL debut.
"I thought he played very well for his first game," Sabres coach Ted Nolan said. "He can be very proud of his first experience."
Follow David Satriano on Twitter: @davidsatriano