New York will finish third in the Metropolitan Division and play the Carolina Hurricanes in the Eastern Conference First Round.
"It's been a bit of up and down, but the last few weeks we find ourselves in a spot where we can kind of put it in our own hands, and we string together a nice few weeks to get ourselves in," Nelson said. "I think this group and everyone here with the team believes in what we have here and what we're capable of and thinks we can get it done."
Timo Meier scored, and Jake Allen made 15 saves for the Devils (38-39-5), who lost four of their final five regular-season games (1-3-1).
"I hope that everybody remembers how it feels right now," New Jersey captain Nico Hischier said. "I think it [stinks]. It's just a little emptiness in there, but you can take things out of this year. Everything is easier when you're winning, but if not, that's where you can build character. For our group, we still have a lot of to learn. We got hit a couple of times in the face this year, went through some adversity, but I think if we take things the right way, we can bounce back with a strong effort next year."
Jean-Gabriel Pageau gave the Islanders a 1-0 lead at 12:22 of the first period, scoring with a one-timer from the left face-off circle off a pass from Engvall on a 2-on-1.
New Jersey allowed the first goal in 57 of its 82 games this season.
"Talent doesn't always bring you to the playoffs," Devils forward Jesper Bratt said. "I think we're missing out sometimes ... playing a little immature, playing a little too fancy. We can't really count on it in the hard games area and we need to come out ready every game to play instead of having to trail from behind two-thirds of the games."
Palmieri made it 2-0 with a power-play goal at 16:52 of the first, redirecting a point shot from Mike Reilly.
Meier cut it to 2-1 at 3:25 of the second period. Bratt's initial shot went in and out of the glove of Varlamov, and Meier scored on the rebound.
Nelson pushed it to 3-1 at 11:48 when he picked up a loose puck in the slot and scored with a wrist shot.
"There's a lot of things I love about our game," Roy said. "I love how we move the puck offensively, I love how we defend defensively. I mean, I love the way we forecheck and the way we played together. Our guys are fun to watch, and I'm sure our fans are pretty proud of our guys right now."