Nicolas Jackson put the home side ahead early on at Stamford Bridge, but Alexander Isak's brilliant finish levelled matters just before half time.
Cole Palmer's powerful strike put the hosts back in front after the restart and Mykhailo Mudryk made it 3-1 soon after to leave Eddie Howe's charges with a difficult task ahead of them.
And while Jacob Murphy's thumping effort reduced the arrears late on, it proved just too late to mount a fightback as the Magpies left London empty-handed.
Jackson's neat flick with barely five minutes on the clock handed the Blues the lead; it was a delicate touch from the forward, who diverted Palmer's trickling effort from outside the box into the corner of Martin Dúbravka's goal.
In a low-key first period, chances were at a premium, although Dúbravka was just about the busier of the goalkeepers. He got down well to keep out a Jackson drive from distance but Newcastle's cause wasn't helped by the withdrawal of the injured Anthony Gordon, who was replaced by Murphy in the half's latter stages.
It was a blow for the visitors but they responded to it superbly. Joe Willock sent a Miguel Almirón cross over the top after Murphy's centre had caused problems, and just before the break they levelled the scores. It was an excellent finish from Isak, who took a touch to control Bruno Guimarães' pass before whipping his shot into the far corner.
Jackson had the ball in the net in added time after a run from deep was denied by the offside flag, while either side of the break Raheem Sterling tried his luck, with his first attempt saved by United's Slovakian stopper and his second flashing past the upright. But Chelsea kept pushing and, after a bright start to the second period, they moved back in front. This time it was Palmer who got it, rifling home from the edge of the box after being picked out in space in front of the Magpies' backline. His low shot was unerring and left Dúbravka with no chance.
It could have got even worse for Newcastle almost straight after they fell behind when Sterling broke free after latching on to a ball over the top. He delayed his finish and rounded Dúbravka but, with a pass to the unmarked Palmer seemingly a better option, he opted to shoot himself and found the retreating Dan Burn in his way as his shot was blocked on the line.
But with quarter of an hour remaining, the hosts did find a third. It was well-taken by substitute Mudryk, who picked up possession from Conor Gallagher and evaded Fabian Schär's challenge before rounding Dúbravka and finding the net despite the best efforts of Emil Krafth on the line.
A tempting delivery from Murphy almost found Isak at the other end and just before the 90 minutes were up, Murphy produced the goal of the game with a rasping drive which thundered into the top right-hand corner.
But in truth, this was a disappointing evening for United, who will now turn their attentions to Saturday evening's Emirates FA Cup quarter-final at Manchester City.
