Jonjo Shelvey swept Eddie Howe's charges into an early lead on Merseyside after Ryan Fraser's cross had been inadvertently turned into his path by Thiago.
But the Reds hit back, with Diogo Jota poking home a controversial leveller before Mohamed Salah fired home, punishing a mistake from the Magpies.
Newcastle fought hard and could have had a penalty when Trent Alexander-Arnold appeared to fell Fraser in the second half, but in the end it was a moment of quality from the Liverpool full back three minutes from time which made sure of the victory for the hosts.
Predictably, the in-form Reds – winners of their last seven in all competitions – started with intent. Salah's backheel found Jota, who turned and forced a smart save from Martin Dúbravka.
But a minute later, in fine fashion, United stole an early advantage. Allan Saint-Maximin and Fraser led the breakaway, with the latter's cross troubling Thiago. The Spaniard didn't connect well with his attempted clearance and former Liverpool man Shelvey picked up the pieces. His curling 20-yarder was aimed to perfection, bending away from Alisson and into the bottom left-hand corner.
It was a terrific opener but Klopp’s men reasserted their dominance quickly. Sadio Mane rattled the upright not long before the visitors lost Jamal Lewis to injury. Ibrahima Konate nodded past the post and Jota fired wide, but their leveller, which didn’t take long to arrive, was a hotly-contested one.
It came through Jota but, with Newcastle’s Isaac Hayden down holding his head after a collision in the box, referee Mike Dean allowed play to continue. The Portuguese frontman's first attempt, a point-blank header, was well kept out by Dúbravka, but his second effort found the roof of the net, with United's protestations doing little to move the officials.
Almost immediately, though, another mistake from Thiago almost let them in for an equaliser. His pass was wayward and Saint-Maximin collected it, driving forward before blasting goalwards, but Alisson got down well to parry it away.
The chance seemed to jolt the Reds and 25 minutes in, they took the lead. It came after an error from Shelvey, whose loose pass back was intercepted by Mane. The former Southampton forward did well to ride the challenge of Fabian Schär, who may well have been penalised for a foul had Salah not tucked away the rebound after Dúbravka kept out Mane’s first effort.
There were other openings before the break, most notably for Salah who rolled one wide, but the early exchanges after the restart were a little less lively. Dúbravka did well to hold on to a Mane header across goal after a well-worked set piece and at the other end, Jacob Murphy cut inside and tried it luck, with the ball whistling past Alisson's post.
The Magpies were a threat on the break, just as they had been in the first period, and could have had a penalty when England full back Alexander-Arnold clumsily tackled Fraser as he looked to race onto Murphy’s low cross. He didn’t appear to get much of the ball, if any, but again Dean was unmoved.
Shelvey then whipped a free kick just wide of the target as Howe's side dug in with leading goalscorer Callum Wilson introduced from the bench in place of the injured Saint-Maximin for the final 12 minutes, though Dúbravka was still the busiest of the two keepers. He scooped away Mane's shunted shot, saved from Jota with his feet and then clutched substitute Naby Keita's audacious chip.
But there was little he could do about Alexander-Arnold's 87th-minute rocket. From around 25 yards, his strike was fierce and flew past the Slovakian into the roof of the net, sealing the points for the hosts and ensuring that United’s wait for an away win would continue.
