Marco Silva's side - reeling from their EFL Cup exit - carved out several openings against their well-rested foes, but they were made to pay for absent ruthlessness as Sean Longstaff and Dan Burn netted either side of the half-time whistle for the Magpies.
Amid ongoing speculation over a move to the Saudi Pro League, Miguel Almiron's omission from the Newcastle squad raised several eyebrows, although the Magpies soon clarified that the Paraguayan was missing through illness.
Almiron's replacement - the returning Jacob Murphy - appropriately had the first opening of the contest with just six minutes played as Fulham made a mess of playing out from the back, but the winger fired straight into the grateful grasp of Marek Rodak.
The fresh-legged visitors continued to come forward in droves, and both Kieran Trippier and Anthony Gordon came close to drawing first blood in the 14th minute; the former's cross was hooked clear by Rodak before Gordon fired wide.
Having weathered that early storm, Fulham started to show a few signs of life in the final third and tested Martin Dubravka for the first time in the 18th minute, but he was equal to Rodrigo Muniz's low strike before gratefully watching the Brazilian's 33rd-minute strike fly a whisker wide via a nick off of Sven Botman.
However, the hosts' improvement was rendered meaningless in the 39th minute, as an episode of pinball inside the Fulham box ended with Longstaff lashing a half volley into the bottom corner, and despite replays appearing to show that the ball had deflected into the midfielder's path off of Bruno Guimaraes's arm, the goal was allowed to stand.
Eddie Howe's men had the ball in the back of the Fulham net once again in the 45th minute, as Murphy fired a venomous strike across goal and in via the post, but the offside flag belatedly, albeit correctly, went up.
The Newcastle winger would have felt a sense of empathy for opposite number Harry Wilson only seconds later, as the Welshman succumbed to a shoulder injury sustained earlier in the first half, being replaced by Willian as three minutes of added time went up on the official's board.
Only three minutes of the second half had passed before Willian sent a ferocious strike past Dubravka, but the Brazilian was also denied by the linesman's flag, and Newcastle soon made the most of that reprieve.
On the 61-minute mark, Rodak made a strong save from a bullet Botman header, but the Fulham shot-stopper could not pounce on the loose ball quicker than Burn, who crashed home the visitors' second against his former club.
Howe's men could arguably count themselves fortunate to be two goals ahead, but the Magpies succeeded in sucking the life out of Fulham with their insurance goal, and the Cottagers' next chance did not arrive until the 77th minute; Raul Jimenez sent a header agonisingly over the bar.
Antonee Robinson fired into the side netting just one moment later, but Fulham continued to face a brick wall of Newcastle shirts and resorted to sending in hopeful crosses, which inevitably ended with the hosts losing the aerial contests.
Captain Tom Cairney was presented with a golden chance to halve the deficit in the fifth minute of added time, but as the midfielder opened up his body in a central position, his swerving side-footed effort missed the mark to preserve Fulham's clean sheet.
After learning their fifth-round opponents in Sunday's draw, Newcastle will return to Premier League action away to Aston Villa on January 30, while Fulham host Everton on the same evening.