Grabner scored on a breakaway 3:53 into overtime as the Islanders defeated the Winnipeg Jets 3-2 on Tuesday.
The victory came just hours after defenceman Andrew MacDonald was traded to Philadelphia and winger Thomas Vanek's future was up in the air.
Cal Clutterbuck sent a pass from around the blue line up to Grabner, who slipped between Jets defencemen Jacob Trouba and Zach Bogosian and beat Ondrej Pavelec with a low shot.
The Islanders (24-32-8) also got a goal and assist from Anders Lee and a goal from Colin McDonald to end a two-game losing skid.
Jets captain Andrew Ladd scored his 16th and 17th goals of the season, tying the game 2-2 with 5:07 left in the third period for Winnipeg (30-26-7), which had a two-game winning streak end.
Earlier in the day, the veteran MacDonald was traded to Philadelphia for centre Matt Mangene and a third-round pick in this year's NHL draft and second-round pick in 2015.
Vanek, who has 21 goals and 53 points, was then a healthy scratch amid rumours he'll be traded before Wednesday afternoon's trade deadline.
"I don't really try to think about it," Grabner said of the day's turmoil. "I've been traded before, I've been on waivers. It's business. You can't really influence what happens...You have to just stay in the moment, day by day, and after (Wednesday) we'll be all done for at least a couple of months again.
"I've been through this, I've seen people traded on game days. Of course, it's weird. (Vanek) came to the rink with us and then he's told he's not playing. It changes things around. But like I said, we did a great job of focusing on not letting this stuff distract us."
New York captain John Tavares is already out for the rest of the season with a knee injury he suffered playing for Canada in the Olympic Games.
"It was difficult," Isles coach Jack Capuano said. "I've been around Andrew a long time and watched him grow in this league. He's been a great player for us, a great team guy and a great friend to a lot of guys, so it's tough the first game without him.
"And, obviously, Thomas has been with us since the start of the year. We all know it's a business, and every guy will tell you that we wish him the best of luck, both of those guys."
Ladd scored his first goal after Islanders forward Josh Bailey turned over the puck in the Jets' zone under pressure from defenceman Mark Stuart.
Winnipeg centre Olli Jokinen then sent a pass up the ice to Ladd, who went in alone on goalie Nilsson and beat him with a low shot on the blocker side at 14:53.
"I thought we came out tight, but we battled back and got a lot of chances and found a way to get it to overtime," Ladd said.
"I think we'll take the positive of getting one point and move on."
In the final three minutes, Nilsson was called for delay of game when he shot the puck over the glass, but the Jets couldn't capitalize on their fifth power play of the game.
Nilsson made 36 saves and Pavelec stopped 24 shots.
The Jets, now 11-3-2 under head coach Paul Maurice, who took over after Claude Noel was fired Jan. 12, didn't help or hurt themselves too much in the playoff race.
Winnipeg is now one point back of the final wild-card spot in the Western Conference, which is held by the Dallas Stars, who lost 4-2 Tuesday in Columbus but have a game in hand.
Winnipeg sits at 67 points, one ahead of Vancouver, which fell to Phoenix 1-0 Tuesday night.
"They were a tough opponent tonight," Jets winger Blake Wheeler said. "And like I said, we're proud of the fact that we were able to rally and get a point."
Winnipeg opened the scoring when centre Bryan Little won a faceoff against Frans Nielsen and pushed the puck ahead to Ladd, who went to the front of the net and beat Nilsson between the pads at 10:57.
The Islanders, who were dominating the first period, kept pressuring Pavelec and got a lucky bounce when Ryan Strome's shot went off the backboards and out onto McDonald's stick and he fired the puck in at 17:30 to tie the game.
New York scored 1:33 into the second period on a goal that was reviewed for high sticking.
Lee went in alone on Pavelec, who got a piece of his shot, but the puck went high behind him and Lee put his stick up and knocked it down and into the net for his third goal of the season.
The Jets outshot the Islanders 17-7 in the middle frame to up their margin 26-20, but they lost rookie Mark Scheifele to a knee injury.
The young centre took a hit from defenceman Calvin de Haan with just under nine minutes left in the period, fell to the ice and went straight to the dressing room.
Maurice said Scheifele, who has 13 goals and 34 points this season, will be evaluated on Wednesday.
Winnipeg hosts Los Angeles on Thursday.
Notes: Tuesday marked Paul Maurice's 1,100th game as a head coach.