Brock Nelson scored and had the only goal in the shootout for the Islanders (26-10-4), who have won four in a row. Ilya Sorokin made 25 saves and stopped all five shootout attempts.
"We came out pretty good and got the early lead, something we've been talking about and want to do more of," Nelson said. "But I think tonight, we maybe took our foot off a bit, they had some chances. The five-minute power play (in the first period), we really didn't get much on it for momentum. You'd like to get something out of that. They were able to kind of find one, get a weird one and get back into it, and then the second period they probably outplayed us. Some nights it's not always going to go your way, it's going to be tough. It's nice that we're in here now on top."
Palmieri had two shots on goal in 15:11 of ice time, and Zajac had one shot in 12:34 the day after being traded by the New Jersey Devils. They played on a line with center Jean-Gabriel Pageau.
"Obviously, in the last 24 hours there was a lot of emotion," Palmieri said. "A lot going on, but I got to meet everybody this morning and everyone was so welcoming. To be a part of a team like that that cares about each other and plays for each other, it was evident the second I walked in the locker room. That's exciting for any player to be a part of."
Nicolas Aube-Kubel and Jakub Voracek scored for the Flyers (18-15-6), who have lost four of their past five games (1-2-2). Carter Hart made 21 saves.
"It is frustrating. We've got to find [a way] to win those games," Voracek said. "You have no other choice [but to] kind of stay focused. Obviously, if you get four, five wins in a row, you are right in the mix. That's keeping us alive. Hopefully we are going to put it together. We have to. We have no other choice."