Aiming to advance into the fifth round of the prestigious competition for the second time in their last 18 attempts, Sean Longstaff's rifled strike sent the Magpies in front before the half-time whistle at Craven Cottage.
Dan Burn's close-range finish, against his former club, extended the visitors' advantage shortly after the hour mark before Eddie Howe's side held firm to record back-to-back away wins in all competitions for the first time since April.
Jacob Murphy, handed his first start since suffering a recurring shoulder injury against Arsenal in November, almost sent the visitors into an early lead after capitalising on a defensive mix-up but his attempt at goal was blocked by Fulham goalkeeper Marek Rodák.
United, targeting a fourth consecutive away win against Fulham, created another opportunity to break the deadlock shortly after as Kieran Trippier's cross from the left flank flew dangerously into the penalty area, forcing Rodák into palming the ball away from the six-yard area before Anthony Gordon's low drive from the edge of the box flashed wide of the left-hand post.
Newcastle goalkeeper Martin Dúbravka produced an important save to prevent Rodrigo Muniz from opening the scoring for the hosts as the Slovakian shot-stopper sprung down low to push away the Brazilian's first-time effort at goal.
Muniz continued to cause a threat to the United backline as the South American forward collected Bobby De Cordova Reid's pass inside the box before his resulting shot was crucially deflected out for a corner by Sven Botman.
In the 39th minute, Newcastle took the lead through Longstaff's emphatic finish, the midfielder's fourth goal of the season in all competitions for the Magpies.
After the Fulham defence failed to clear Trippier's in-swinging cross, the ball eventually fell kindly into Longstaff's path, who clinically lashed the ball into the bottom right-hand corner, with the goal standing despite the hosts protesting for an adjudged handball from Bruno Guimarães in the build-up.
Murphy put the ball in the net on the stroke of half-time for Newcastle after racing onto Alexander Isak's through ball before cooly drilling his resulting effort into the left-hand corner but the Wembley-born winger's strike was disallowed having strayed into an offside position.
In the second-half, Newcastle pushed for their second goal of the evening as Isak, who had netted twice in the Magpies' 3-0 away win against local rivals Sunderland in the Emirates FA Cup third round, took aim from the edge of the box but the Swedish forward's effort from the edge of the box was held by Rodak.
Burn, who had scored against his former club in December's 3-0 win at St. James' Park, doubled the Magpies' advantage in the 61st minute. After Botman's header from Trippier's corner was parried by Rodák, the Blyth-born defender was quickly on hand to place the ball home from close-range.
Fulham went close to halving the deficit as Raúl Jiménez, who was sent off in last month's Premier League encounter against the Magpies at St. James' Park, rose above Botman from Kenny Tete's lofted cross but the Mexican's header was directed over the crossbar.
Newcastle survived a further attack from the West London outfit as Tete's cross from the right wing found its way to Antonee Robinson but the American left-back sliced his attempt into the side-netting.
Backed by 3,800 away fans, including the Magpies' all-time leading goalscorer Alan Shearer, the visitors preserved their clean sheet in the final stages and maintained their hopes of winning a seventh FA Cup title.
