Martin made it 2-0 at 4:48 of the second period, scoring on a snap shot from the right circle following a turnover in the neutral zone by Jacob Trouba. The goal was Martin's third of the season and first since Feb. 9 (24 games).
The Rangers outshot the Islanders 19-12 in the first two periods.
"A little bit of everything," Rangers forward Mika Zibanejad said about what was missing. "Intensity, just playing together, I think, pretty much the way we've been playing all year. Play fast and we're on the same page. We weren't tonight.
"I wish I had the answer. If I had the answer, I don't think it would look like this, obviously. … It's just not acceptable. Obviously, wasn't good enough."
Barzal extended the lead to 3-0 at 2:47 of the third period. After the puck slipped away from Schneider at the Islanders blue line, Oliver Wahlstrom sped through the neutral zone and gained possession before passing to Barzal, who skated in for a backhand.
"Any player that comes into [Madison Square Garden] gets a little more fired up," Barzal said. "It's a fun atmosphere. Aside from that, I think it's just Islander-Ranger rivalry. We want our fans at the end of the day to celebrate after games against them. I think that just fires me and our team up a little bit more."