In what was a repeat of last year's final, the visitors proved too strong for a hugely disappointing Man United, who were unable to bounce back from their three-goal loss in the Manchester derby.
Miguel Almiron and Lewis Hall scored for the visitors in the first half before Joe Willock netted in the second, with Eddie Howe's side fully deserving of their position in the final eight.
Both managers made wholesale changes from their weekend fixtures in the Premier League, with Man United's team showing seven changes from the Manchester derby, while there were eight changes to the Newcastle side that started the 2-2 draw with Wolverhampton Wanderers.
Th visitors lost Matt Targett to what looked to be a serious hamstring injury in the early exchanges, and it was Man United that made the better start, with the Red Devils moving the ball with pace and purpose.
Newcastle goalkeeper Martin Dubravka, who was on loan at Man United in the first half of last season, had to make a smart save to keep out a low drive from Casemiro in the 13th minute.
Hannibal Mejbri picked up an early booking and was fortunate not to have another yellow shortly after, with the midfielder flying into another challenge.
Matt Ritchie, starting for the first time since January, sent a powerful attempt wide of Andre Onana's goal in the 22nd minute, with Newcastle just starting to work their way through the Man United midfield.
The Magpies made the breakthrough in the 28th minute of the match, and it came courtesy of excellent work from Tino Livramento, who robbed possession off Alejandro Garnacho down the left before driving forward and releasing Almiron, and the substitute took a touch before finding the far corner of the net.
Mount turned a low Garnacho cross wide of the post, before Martial just got under a cross from Diogo Dalot, as Man United looked for a quick response.
Sergio Reguilon then fired one over the crossbar in the 35th minute, but Newcastle doubled their lead less than a minute later, with Hall picking out the bottom corner with an excellent volley.
The Magpies again got in down their left, and it eventually broke for Hall just inside the penalty box, with the youngster expertly placing it beyond the diving Onana.
Willock fired into the body of Onana in the 41st minute, before Dalot tried his luck down the other end with a strike that whistled past Dubravka's goal.
Dubravka then had to fly across his goal to keep out a powerful drive from Mount, with Newcastle resisting some pressure to keep hold of their two-goal advantage at the break.
Aaron Wan-Bissaka and Sofyan Amrabat entered the field for Dalot and Casemiro, and the hosts had an early chance in the second period, with Martial turning a low cross wide of the near post.
Amrabat then curled one over the crossbar in the 54th minute, before Antony smashed one over two minutes later, with Man United showing more purpose after the break.
However, Newcastle took the game away from the home side on the hour, with Willock driving towards the Man United penalty box before smashing the ball into the bottom corner.
Bruno Fernandes, who entered alongside Rasmus Hojlund and Marcus Rashford, missed the goal from a narrow angle in the 74th minute, before Longstaff came close with a brilliant strike in the 76th minute.
There was nothing for the home fans - those who remained until the end - to cheer, though, with Newcastle breezing into the final eight of the competition that Ten Hag won last term.
Man United will be bidding to bounce back when they head to Fulham in Saturday's early start in the Premier League, while Newcastle will take on Arsenal later in the afternoon.