Alexander Isak's penalty had put the Magpies ahead at St. James' Park, but Michail Antonio's leveller and Mohammed Kudus' drive handed the Hammers a half time lead.
Jarrod Bowen then made it 3-1 just after the restart and it looked as though Newcastle - who lost Jamaal Lascelles, Tino Livramento and Miguel Almirón to injury - were beaten.
But Isak's second penalty lifted spirits on Tyneside and the forward's superb through ball for Barnes' well-taken first drew Eddie Howe's men level inside the final ten minutes.
And the winger's second in the 90th minute - a wonderful strike from range - completely an incredible turnaround, sending St. James' wild in the process, with the stoppage time dismissal of Anthony Gordon doing little to dampen United spirits.
Isak got the Magpies off to the perfect start with a typically cool conversion from the spot. The penalty was awarded after Gordon was bundled over by Vladimir Coufal in the box as the pair stretched for a loose ball and, despite a lengthy VAR delay as officials checked for a potential offside, the Swedish frontman remained calm and sent Alphonse Areola the wrong way to handed Howe's side the lead.
Isak later hooked one over before United were forced into an early reshuffled when captain Lascelles succumbed to injury, with Emil Krafth sent on in his place, and the visitors began to assert themselves. Antonio swiftly turned Dan Burn, who had been shifted inside to centre half after Lascelles' withdrawal, but his snapshot was too high.
It was a warning for the hosts, though, and with 20 minutes gone Antonio did find the target. It came after a delicate lofted pass from Lucas Paqueta which left the Hammers' lone frontman in space, and he duly homed in on goal and slotted the equaliser past Martin Dúbravka.
Gordon glanced a Murphy cross just wide and Dúbravka saved a first-time shot from Bowen as the game became open and increasingly end-to-end in the latter stages of an elongated first half, with Areola holding an angled Sean Longstaff cross-shot. Bruno Guimarães went even closer in a frantic ten minutes of added time, rattling the crossbar from Murphy's cutback. Seconds later, Burn just about halted Kudus as he looked to go one-on-one, before Paqueta looped a header onto the roof of the net.
In the final few minutes of added time, however, the Hammers found a way through once more. It was Kudus who applied the finish, with Dúbravka getting a hand to his effort but seeing it agonisingly drop down into the far corner. The Magpies were aggrieved that Kudus appeared to have caught Fabian Schär with a flailing arm in the build-up but referee Rob Jones opted not to halt play, leaving Newcastle with work to do in the second half.
The Hammers, who replaced goalkeeper Areola with Lukasz Fabiankski at the break, took just over two minutes to move further in fromt. It came from a swift counter-attack after a Newcastle corner, with Kudus skipping away from Schär on the left and finding Bowen free in the middle. The England international sprinted clear before planting a low right-footed finish beyond Dúbravka to stun St. James' Park.
The hosts responded by sending on Elliot Anderson, Almirón and Lewis Hall, the latter in place of Livramento, who hobbled off. The switches preceded some pressure, with West Ham sitting deep and attempting to stifle them, but clear opportunities were hard to come by. One fell to Isak, who scuffed his shot well wide in a congested penalty box, just before another almost identical opening presented itself to Longstaff, with the outcome the same.
Newcastle's injury nightmare worsened further as they lost a third player, the substitute Almirón, with a quarter of the contest left, but they were handed a lifeline when Gordon went down under Kalvin Phillips' challenge in the box with quarter of an hour left. Another VAR check, and with referee Jones having consulted his pitchside monitor, led to the award of the spot kick and Isak duly dispatched once more, raising the volume inside the ground.
And with eight minutes left, Isak's quality helped haul his side level. Picking the ball up in the centre circle, his through ball was perfect for substitute Barnes, who had timed his run brilliantly. The former Leicester man's finish was unerring, low between Fabianski's legs, and all of a sudden Newcastle were playing for the win.
They almost got it when Longstaff's clipped ball across the face of goal flashed beyond Fabianski but Barnes, flying in at the far post, just couldn't quite connect with it at point-blank range. Moments later, however, in the final minute of the 90, they did get it. It was Barnes who got it, wrapping up a sensational comeback with a powerful curling strike that nestled beautifully in the bottom corner.
There was still time for Gordon to pick up his second yellow card, ruling him out of Tuesday's clash with his former club Everton, but it mattered little in the moment. St. James' Park hasn't seen too many occasions like that in the Premier League during 2023/24; this was one to savour for those in black and white.
