St James' Park erupted on 55 minutes when Ben White's own goal gave the hosts the lead.
Bruno Guimarães doubled The Magpies' lead five minutes from time when he slotted into an empty net.
The defeat for Mikel Arteta's men means that they need their North London rivals Tottenham Hotspur to lose to an already relegated Norwich City on the final day, while they must grab maximum points at home to Everton.
Story of the game
After the 5-0 drubbing at Manchester City last time out, Eddie Howe made two changes to his Newcastle side, bringing in Callum Wilson from the off for his first start since late December, as well as Fabian Schar.
Mikel Arteta was sweating on the fitness of centre-back duo Gabriel and Ben White heading into this match of monumental importance to The Gunners.
Despite one being forced off with an injury in the 3-0 loss away bitter rivals Spurs – that could prove pivotal in Arsenal's season – and the other not fit enough to feature at all in midweek, the duo both started on Tyneside this evening.
Once the stadium-wide flag display concluded with a side dose of Rockin' All Over The World by Status Quo being blasted over the PA system – which had both sets of supporters just as happy for obvious reasons – the game got underway, and the visitors did not settle well.
Their poor start didn't let up, as just seconds later, White was shown a booking for a foul on Wilson just outside of the box.
White versus Wilson was turning into quite the battle early on.
The England defender caught the forward offside a handful of times in the opening 15 minutes. But he was forced into more evasive action shortly after when he cleared the No. 9's header off the goal line.
In an opening exchange dominated by the hosts – highlighted by them having more than 60 percent possession – Bukayo Saka was the singular shining light in the Arsenal side.
Within the space of three minutes, he alerted the Newcastle defence to the quality he has when he cut in from the right-hand side and fired at Martin Dubravka's goal, but his effort was saved by The Toon goalkeeper
Wilson had something to show for his on-pitch battle just before the half-an-hour mark, when pictures revealed his front tooth was hanging out after a challenge.
It didn't rule him out of the rest of the game, but it certainly would rule him out of smiling in the customary team photo if Newcastle were to go on and win the game.
The last big chance of the half did fall to the hosts when Allan Saint-Maximin cut in from the left-hand side and skinned Takehiro Tomiyasu – who was then forced off with an injury moments later – before firing low and hard, but Ramsdale was equal to it, making a superb stop.
Toon take all the points with two second half goals
Moments after the restart, Newcastle were forced to make a concussion substitution after Schar was knocked out cold following a collision with Eddie Nketiah.
The Swiss international eventually got up, much to the delight of everyone inside the stadium, but could not continue.
In a game that Arsenal simply had to win to keep their Champions League dreams alive, they fell behind on 55 minutes.
Joelinton made a lung-busting run down the left-hand side before crossing low into Wilson. White got a toe on it, but diverted the ball into his own net.
Wilson came close to doubling Newcastle's lead moments after that Arsenal change, but his volley narrowly went over Ramsdale's crossbar.
Mohamed Elneny looked to pull The Gunners level when he took aim from range on 71 minutes, but his effort was tame and comfortably caught by Dubravka.
With 15 minutes to go and the visitors needing to win the game, they made more offensive changes, with Gabriel replaced by Nicholas Pepe.
But it left them incredibly frail defensively, and Wilson was keen to take full advantage of that.
From well over 40-yards out, he chested the ball before audaciously at goal – but fortunately for a beaten Ramsdale, Wilson's effort crept just past the post.
But there was nothing he could do to stop Guimarães from doubling Newcastle's lead, and inadvertently sending Arsenal's harshest rivals one giant step towards the top four.
Sean Longstaff played a ball through to Wilson, Ramsdale did his best to parry the ball to safety, but it fell to Guimarães who was in the perfect position to slot the ball into an empty net.
And just like that, The Gunners' Champions League hopes look to have slipped through their fingers.
But they were deservedly beaten by a hungrier and simply better Newcastle tonight – and they only have themselves to blame if they eventually do fall short of the top four...
Newcastle: Dubravka; Krafth, Schar, Burn, Targett; Guimaraes, Longstaff, Joelinton; Almiron, Wilson, Saint-Maximin. (4-3-3)
Arsenal: Ramsdale; Tomiyasu, White, Gabriel, Tavares; Xhaka, Elneny; Saka, Odegaard, Smith Rowe; Nketiah. (4-2-3-1)