The hosts were left frustrated by a well-drilled Sunderland side in the first-half, going close a number of times but failing to ask much of goalkeeper Jordan Pickford.
Their cause was not helped by injury to Philippe Coutinho, who had to be stretchered off with an apparent ankle problem following a coming together with Dider Ndong on 33 minutes.
The Reds were denied time and time again in the second-half, Roberto Firmino, Sadio Mane and Emre Can all going close.
But with it looking like they could be left rueing their missed opportunities, Coutinho's replacement Divock Origi curled into the far corner to break Sunderland's resistance.
And Liverpool sealed the three points to temporarily send them back to the summit of the league table on goal difference thanks to James Milner's late penalty, after Mane had been brought down by Ndong.
Liverpool are now 14 games without defeat at home in the league, although Chelsea's win over Tottenham Hotspur in Saturday's evening kick-off saw them once again leapfrog the Merseyside outfit back to the top.
Liverpool unchanged as Sunderland make two changes
Jürgen Klopp named an unchanged team for the first time in 41 league matches, a run stretching back to the German's second game in charge of the club.
With Adam Lallana still absent from a groin injury sustained over the international break, Georginio Wijnaldum kept his place in midfield alongside Jordan Henderson and Emre Can.
The only notable absentee was Daniel Sturridge, who was ruled out with a slight calf strain, allowing 17-year-old Academy starlet Ben Woodburn to feature on the bench for the first time.
David Moyes' Sunderland were forced to make two changes from the side that thrashed Hull City 3-0 last time out, Papy Djilobodji having been sent off and Paddy McNair picking up a season-ending knee injury.
Captain John O'Shea returned to the team in central defence, while Steven Pienaar also came back in from suspension as Sunderland searched for a third straight league victory for the first time since May 2014.
Before the game, Moyes was pretty frank in his admission that the Black Cats would need to park a double-decker bus in front of goal to keep out a Liverpool side that had plundered 17 goals in just five previous home league games.
The Scot proved true to his word, the visitors consistently putting numbers behind the ball from the early stages and starving Liverpool of space in the final third.
Firmino and Wijnaldum both got shots away on goal in the opening 15 minutes, but Pickford saved both comfortably as Liverpool struggled to disturb the visitors' back-line.
Sunderland defend in their numbers to keep Reds out
Sunderland - boosted in confidence by their back-to-back victories - occasionally looked dangerous themselves. Loris Karius did well to block Pienaar's attempt from close-range after hesitating to come and meet a hooked Billy Jones ball back into the box.
Those nerves were commonplace in the opening half-an-hour, perhaps best summed up by one particular instance where a Karius goal kick went straight out for a corner.
Liverpool lacked the kind of fizz and verve of previous home performances, having thrashed Watford 6-1 most recently. Their fluency was done no favours when Coutinho had to be stretchered off with his right leg in a brace following a clattering collision with Ndong.
Liverpool had their best chance for the game's opening goal soon after, after Coutinho's replacement Origi won a free-kick. Henderson's delivery found the unmarked Dejan Lovren at the back post but from seven yards, but the Croatian's header flew across the face of goal and agonisingly wide.
That aptly summed up the frustrations of the first-half from the hosts' perspective, as they prompted and probed without success against a stubborn and resilient Sunderland back-line.
Despite a flurry of Liverpool chances in the final few minutes of the first-half, Pickford remained relatively untroubled. Lovren fired a powerful drive wide from 30-yards, before the Black Cats' goalkeeper kept out Sadio Mane's attempt from inside a packed penalty box.
Liverpool continue to dominate the ball
Klopp's side had enjoyed 80 per-cent of the possession in the first period and continued to dominate proceedings upon the restart, but a simple lack of sharpness often let them down.
A heavy first touch from Nathaniel Clyne saw him squander a good opportunity at the near post, before Mane scuffed a shot into Pickford's grasp with options to his left at the edge of the box.
But Liverpool maintained a constant pressure, Wijnaldum the next to go close as he volleyed Can's flick narrowly wide of the far post before Firmino forced Pickford to dive down low to his left and prevent his effort sneaking in at the near post.
The hosts inched closer with every attempt, Can's half-volley again going incredibly close after the German had ran on to Jones' headed clearance.
Liverpool's desperation for a breakthrough goal left space at the back for Sunderland to break and Karius was required to make a quick intervention to close down Duncan Watmore after being found in space in the box by Anichebe, before the latter's effort was blocked on the follow-up.
It was always the hosts who looked the more likely of finding that first goal, Firmino's low cross somehow evading his teammates before Henderson put a 20-yard free-kick over the crossbar and Mane saw a shot from close-ranged blocked by Jason Denayer.
Late show sees Liverpool claim full three points
But their efforts were without reward until the 76th minute, when Anfield erupted in relief as Origi finally found a way past Pickford to burst the dam with his first league goal since April.
The Belgian striker cut onto his right foot and curled an effort around his marker and into the far bottom corner, beyond Pickford's grasp, for a deserved opener and his first league goal of the campaign.
Liverpool had chances to extend their lead, Can going close as a shot deflected just wide, before Ndong's foul on Mane gifted Milner the chance to kill the game from the penalty spot. That he did, sending Pickford the wrong way to make sure of victory.
In the final minutes, the win was crowned by the introduction of teenager Ben Woodburn, who became the club's second-youngest player ever to play in the Premier League behind Jack Robinson.