The Magpies - led by Jason Tindall and Graeme Jones after head coach Eddie Howe was admitted to hospital on Friday - led through Sandro Tonali's volley, but Alejandro Garnacho levelled for Ruben Amorim's men before half time.
The equaliser proved only a minor setback for United, who had dominated the first period, but there could be no doubting who had the upper hand after the break as Barnes' brilliance brought those in black and white back to the fore.
The 27-year-old's two strikes put the hosts in complete command on Tyneside before Bruno Guimarães rolled home a fourth to add further gloss to a terrific victory - and one which will surely have provided the absent Howe with some much-needed tonic.
The hosts actually struck inside 48 seconds, and it looked like becoming a long afternoon for stand-in Red Devils goalkeeper Altay Bayındır, who had to pick Joelinton's close-range finish out of his own net inside that first minute. Fortuitously for the Turkish stopper and his struggling side, Kieran Trippier - who supplied the cross - was offside.
The in-form Jacob Murphy fizzed one just past the upright as the Magpies provided much of the early ugency. The visitors' first foray forward was a slick one, though, with Bruno Fernandes and Joshua Zirkzee exchanging passes to allow the latter to get a shot away, which Nick Pope did well to push over.
Alexander Isak, seeking his 25th goal of the campaign, glanced a Guimarães cross just wide before playing an instrumental role in a beautifully-crafted opener. His first touch to flick the ball up was perhaps unintentional but his second, a well-weighted lofted assist for Tonali, was perfect, and the Italian's low volley was unerring.
The Swedish forward almost extended the advantage soon after. Tino Livramento's centre was palmed out by Bayındır, who then did excellently to recover and get a hand to Isak's left-footed follow-up before it was scrambled clear. Dan Burn, dominant in the air all afternoon, rose to meet a Harvey Barnes cross but saw his header drop just over.
But in the final eight minutes of the half, Amorim's side suddenly found some rhythm. Their leveller came against the run of play and in part thanks to a driving run from full back Diogo Dalot, who ran at the heart of the United backline before playing in Garnacho. The winger's finish, angled just beyond Pope, nicked the upright on its way in and buoyed those in Red; the goalscorer had another go a few minutes later, but found Pope equal to his curling effort.
Just three minutes after the restart, Newcastle moved in front again. It was a goal of relative simplicity but superb execution. Livramento eased past Dalot and his deep, low cross was kept in by Murphy at the back post. The 30-year-old earned his 11th assist of the campaign, brilliantly turning the ball back across the face of goal, first time, for Barnes to arrive on cue and fire home from four yards out.
The Red Devils supplemented their attack with the introduction of Rasmus Højlund, who replaced the injured Zirkzee, while Mason Mount and Patrick Dorgu also came on but were powerless to halt the Magpies' flow. Burn nodded a Murphy cross, which Barnes had also risen to attack, over but just after the hour the former Leicester City man had his second. It looked like little more than a half-chance when Noussair Mazraoui slipped over in possession but Barnes didn't hesitate, picking up the ball and shifting through the gears. His run took him right through the middle of the Manchester United centre halves and his finish was magnificent, arrowed into the top right-hand corner in front of the Gallowgate End.
The third goal seemed to rob the visitors of what little fight remained in them. The fourth was gifted to Guimarães by Bayındır, who inadvertently played the ball straight to Joelinton. The Brazilian was alert and intercepted the attempted pass, nodding the ball straight to his captain, who rolled home with confidence to wrap things up. The result leaves Amorim's men in 14th in the Premier League and lifts the Magpies into the top four, as their push for Champions League qualification continues apace.
