Martin Dúbravka saved a Mohamed Salah penalty in the first half before the Egyptian put the Merseysiders ahead in the second.
Alexander Isak equalised for the Magpies but, despite Dúbravka's heroics in goal, Curtis Jones, Cody Gakpo and another Salah spot kick wrapped up the three points for the Reds after Sven Botman's header had made it 3-2.
The home side started strongly, with Botman deflecting a Trent Alexander-Arnold strike behind, but United responded with a shot from Lewis Miley - the Magpies' Sela Player of the Month for December - which was blocked by Joe Gomez.
In the 12th minute, Dúbravka made the first of several great saves, this one from Darwin Núñez, before Fabian Schär got in the way of Jones' follow-up.
Núñez headed over from an Alexander-Arnold free kick, then Luis Díaz had a goal ruled out for offside.
But a few seconds later, the Colombian dribbled into the box before tumbling under Botman's challenge - only for Dúbravka to brilliantly keep out Salah's spot kick, and Alexander-Arnold blazed over from the rebound.
With rain falling throughout, Liverpool looked to open the floodgates as Dúbravka got down to hold a low effort from Jones and Joelinton blocked a Van Dijk effort.
In the 36th minute, after Botman had been pushed off the ball by Nüñez, the outstanding Dúbravka saved at the Uruguayan's feet, then parried another effort from the frontman.
The visitors' resilience looked like it had paid off when Dan Burn dived to head home a Miley cross at the far post, but Isak was narrowly offside in the build-up.
An audacious strike from Alexander-Arnold crashed off the woodwork to ensure the scores stayed level at half time, but shortly after the restart, Liverpool took the lead.
A Newcastle attack was halted and the Reds swiftly broke, allowing Díaz to find Núñez, who unselfishly squared for Salah to tap home.
Dúbravka was called into action once more when Salah returned the favour soon afterwards but Núñez was denied at point-blank range, then the Magpies' Slovakian shot-stopper made another fine intervention to frustrate the same player.
Those saves proved all the more important because, in the 54th minute, the Magpies equalised as Liverpool-born Gordon jinked past two defenders and threaded a perfect pass for Isak to run onto. The Swede was unerring in his finish as he momentarily silenced the Kop with his 11th of the campaign.
Dúbravka saved Newcastle again in the 62nd minute, pushing out a Díaz drive, then Núñez sent a header wide of the target.
Salah saw a curling effort patted away by Dúbravka, who then denied substitute Gakpo in the 69th minute.
Gordon had a low shot comfortably fielded by Alisson but then Liverpool took the lead for a second time as Salah and Diogo Jota combined for Jones to net.
A similar move then saw Salah cross with the outside of his boot for Gakpo to make it 3-1, but Botman headed home from Longstaff's right-wing corner to give United hope.
However that was extinguished when Liverpool broke once more and Jota rounded Dúbravka before falling to the ground a couple of steps later.
This time, Salah scored from the spot and United were consigned to a third league loss in a row.
