Substitute Almirón grabbed the first, sweeping home after Tino Livramento's brilliant work in the build-up, before teenager Hall volleyed home a superb second to give Eddie Howe's men breathing space at half time.
The hosts rallied briefly but Willock's terrifically-taken third, on his first start since May, made sure of United's place in the quarter-finals as they impressively disposed of last year's winners, who defeated Newcastle in the 2022/23 final, in the process.
The visitors were dealt an early blow when Matt Targett was forced off within four minutes, with Almirón sent on in his place. It was another worrying development for Howe, who has seen his side depleted with injuries in recent weeks.
Casemiro had an early drive saved by former Red Devils loanee Martin Dúbravka but it was Howe's side who dominated possession early on. On the half hour, Hannibal - who was somewhat fortunate to still be on the pitch, even at that early juncture - played it to Alejandro Garnacho, who was robbed by Livramento. The full back, so impressive on his full debut against Manchester City in round three, burst forward with purpose before sliding a neat through ball for Almirón to run on to. His touch was delicate and his finish, into the far corner beyond Andre Onana, was unerring.
Mason Mount stabbed one wide as the hosts looked for a swift response but it never really looked like coming. Seven minutes later, the Magpies had two, and it was an eyecatching one for Chelsea loanee Hall. The visitors were probing again but there was no hesitation when Harry Maguire's headed clearance dropped into his path, with his volley nestling beautifully in the bottom right-hand corner.
Diogo Dalot whipped a shot just wide before the break and after the restart, having made three changes, Erik ten Hag's men looked bright. Anthony Martial saw his poked effort blocked and one of the substitutes, Sofyan Amrabat, fired over as the Red Devils looked to get on the front foot.
But just after the hour, Newcastle made sure of their win with a magnificent goal from Willock. Having missed six months through injury, the former Arsenal man only returned to the Magpies' matchday squad in recent weeks, but his lack of gametime didn't show as he strode towards the Manchester United box at pace. He made space for himself 20 yards from goal and struck it low, with power and precision, and the ball zipped into the bottom corner past a helpless Onana to seal a memorable away day for Howe's men and their typically vocal travelling support.
It could have got even better for the Magpies soon after when Sean Longstaff - captain for the night on his 150th appearance for the club - whistled a strike just wide of Onana's post, but their place in the last eight was all but confirmed by then.
