The duo produced two impressive goals to put the Red Devils ahead and Sir Alex Ferguson's men held on despite a Papiss Cisse-led fightback in the second half at Old Trafford
The Red Devils were on top for the majority of the first half, and should have took advantage when Anderson and Tom Cleverley both had brilliant chances, but did eventually take the lead courtesy of a superb strike from the Brazilian.
Alan Pardew's men tried to work their way back into the clash after the break, having a couple of decent chances, but Cleverley fired home to extend the hosts' lead. Papiss Cisse pulled a goal back with half-an-hour left, but it wasn't enough to stop his side crashing out.
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The visitors made a number of changes from the side which beat Norwich in the Premier League, with a few unfamiliar faces amongst the starting line-up, though Fabricio Coloccini and Cheick Tiote both returned.
After the home side dominated the opening ten minutes, the Magpies had the first real chance of the game when Dan Gosling volleyed over from six yards out. The midfielder had time and space to pick his spot but sliced the effort well wide after Shola Ameobi's nod down.
Moments later the hosts appealed for a penalty after Rooney was tripped in the area, but nothing was given. Then, they came forward again as Anderson broke from midfield, but the midfielder wasted a great chance by shooting wide.
Ferguson's men continued to play the better football, though Newcastle looked dangerous on the counter and had a number of decent openings. Welbeck, however, was the next to muster a shooting opportunity as he fired a strike towards the corner which Elliot parried away.
Ten minutes before half-time Cleverley made an complete mess of a brilliant opening, from which he should have put his side 1-0 up. After some brilliant play from Chicharito on the right-hand side, and some neat touches from Rooney and Welbeck, the England midfielder was left with a one-on-one with Elliott from no more than three-yards out, but somehow shot wide.
United did take the lead just before the break, though, as Brazilian midfielder Anderson held off a challenge from his man before smashing the ball into the top corner with his left boot from outside the box, via a touch from the post.
Two minutes after half-time the former Porto man had a good chance to bag a second. He started the move with a great ball to Welbeck on the left flank, who then returned the favour, however, Anderson could only lash his first-time strike over the bar.
As Ferguson's men looked to push home their slight dominance, it was the visiting team that had the next good opening as Haris Vuckic smartly worked his way past Michael Keane in the box before curling a strike just past the post.
And they were left to rue that miss just under five minutes later. Welbeck picked up the ball on the edge of the box and shifted it central to Cleverley, who rifled a strike into the corner just out of Elliot's reach.
The two-goal cushion didn't last long, however, as Cisse came off the bench and made an instant impact by rising highest to head home from Shane Ferguson's cross to pull one back.
With a comeback looking on as Newcastle pushed forward, Chicharito almost gave the home side the perfect response, hitting the bar from a narrow angle with a powerful effort, before Anderson thundered a deflected shot not far over the bar.
The Magpies continued to look for an equaliser, throwing numerous balls into the United box for Ameobi and Cisse to attack. The latter also crashed the bar with a fantastic overhead kick after he chested the ball up for himself just inside the box.
Ferguson's men, however, held off other half-chances from the visitors late on, securing the win and marching on to the next round