The Islanders had every opportunity to fumble Tuesday’s game away like they have so many others this season.

But instead of succumbing to the old demons, they proved themselves to be a new team.

One that is looking more and more like it will get a chance to compete in the playoffs.

After going 531 days without playing at UBS Arena, the Rangers left Tuesday without any happy memories, as their comeback attempt fell short against a desperate Islanders side fighting tooth and nail to get into the postseason.

After a 4-2 win over the Blueshirts that teetered on a knife’s edge for the entire final 20 minutes, the Islanders appear to be zeroing in on locking that down.

“Two points,” a sage Cal Clutterbuck said, “is two points.”

With the win, the Islanders stayed in third place in the Metropolitan Division, two points ahead of Washington, which beat the Red Wings to move into a wild-card spot.

With the Flyers getting blown out by Montreal, the Islanders’ lead on Philadelphia moved to four points.

Special teams nearly undid the Islanders, who took three consecutive penalties in the second.

The Rangers scored on two of them, first with Chris Kreider tipping in Artemi Panarin’s shot, then on Adam Fox’s wrister from above the slot that made its way through traffic to cut a 3-0 lead to 3-2.

So, going into the third, no matter what color you were wearing, everyone in UBS Arena was having flashbacks to mid-February at MetLife Stadium.

Those were not ameliorated when the Rangers came in waves to start the final period, establishing a strong forecheck and holding the puck almost without pause.

Despite their early lead, the Islanders were falling into old habits — sitting back, depending on their goaltender and trying to eke out a one-goal win.

“We’d like for it not to be [like that],” said Clutterbuck, who at one point in the third gathered the team for a discussion. “We had to just change the mindset and put a stop to the momentum they generated in the second period with the power plays.”

Like it or not, this was going to be the same combination of Semyon Varlamov and desperate defending that the Islanders relied on Saturday against Nashville.

And aside from a two-minute reprieve after Barclay Goodrow’s slashing penalty, that’s just what this was.

Somehow, for the second game running, it worked, as Varlamov finished the game with 34 saves and the Islanders battled just hard enough to hang on, with Anders Lee’s empty-net goal sealing the deal with 5.8 seconds remaining.

“Most of [the shots] were outside,” Varlamov said. “Our guys were doing an amazing job protecting the middle. That’s what we’ve been trying to do all season long.”

Add in a heavy dose of extracurriculars, with a late uncalled hit from behind from Noah Dobson on Vincent Trocheck and a seemingly accidental elbow by Adam Pelech on Mika Zibanejad drawing descriptions of “vicious” from Rangers coach Peter Laviolette, who also called Pelech’s elbow intentional, and the noise was amped up to 11 ahead of the regular-season finale between these two teams on Saturday afternoon at The Garden.

By night’s end, both coaches were declaring righteousness — both in that spat and in the game itself.

“I thought after 40 minutes, we were the best team on the ice,” Roy said.

“I think we owned the game,” Laviolette said. “The puck was on our stick the entire second and third period.”

That proved a little more stressful than it looked this game would be after the Islanders came out of the gates skating as hard and intensely as they have all season.

Despite Igor Shesterkin denying Mat Barzal’s penalty shot 85 seconds into the game, the Islanders jumped out to a 3-0 lead within the game’s first 13:38, thanks to goals from Mike Reilly, Bo Horvat and Kyle Palmieri — all of which resulted from players getting in front of Shesterkin’s net.

Dobson’s shot that Horvat tipped in marked the defenseman’s 59th assist of the season.

Less than two minutes after that, Dobson notched No. 60, getting the secondary assist on Brock Nelson’s feed to Palmieri in the crease.

That made Dobson the first Islanders defenseman since Denis Potvin to reach 60 assists in a season. And it just so happened that Potvin was watching on in a suite to see him do it.

The franchise record for a defenseman, set by Potvin, is 70.

He did so in 1979, and the Islanders made sure that would be the only evocation of that season on Tuesday by beating the Rangers this time around.