Liverpool earned a third successive away league win at Stoke City for the first time in their history as Mohamed Salah came off the bench to score in his fourth successive Premier League game with a late double.
Sadio Mané put the visitors ahead with a cute lofted finish over Lee Grant on 17 minutes, with Stoke complaining that the ball had gone for a goal-kick before Joe Gomez's cross in the build-up.
The Reds were fortunate that Simon Mignolet was not dismissed for hacking down Mame Biram Diouf on the edge of the box as the last man, with Stoke fans making Martin Atkinson well aware of their thoughts on the matter.
Dominic Solanke and Mané both squandered one-on-ones for Liverpool with the latter hitting the post while Joe Allen twice went close in the second half with the Potters pressing for an equaliser.
But Salah came on in the 66th minute and took his tally for the season to 12 goals in just 14 Premier League appearances with his third brace in four league games.
He first netted a rasping volley from Mané's cross and then applied a clinical finish from Emre Can's forward pass, which Erik Pieters got lost underneath, to continue his sublime form in front of goal.
Contentious Mignolet call mars first-half
After a slow-burning first 15 minutes, Mané ended his longest league drought as a Liverpool player of six games, chipping over Grant from close-range - much to the frustration of the Stoke fans who felt the ball crossed the line for a goal-kick in the build-up.
Gomez seemingly kept the ball on the pitch by mere millimetres on the right and found Solanke with his cut back. The striker controlled well to poke through for Mané, whose effort Grant could not prevent from trickling in.
Stoke were initially subdued but improved after switching to a 4-4-2 set-up with Diouf moving up top to partner Peter Crouch.
But Liverpool should have doubled their advantage before their tactical changes could have their desired impact. Roberto Firmino and Solanke combined to put the latter through on goal - only for the 20-year-old to fire straight at Grant.
Immediately at the other end, referee Atkinson again found himself the subject of several choice chants from the home fans. He issued Mignolet only a yellow card despite the Belgian 'keeper catching Diouf late, as the last man on the edge of the box, on the edge of the area.
And Liverpool wasted another clear-cut opportunity to go 2-0 up when Mané raced beyond Bruno Martins Indi, the Dutchman leaving the goalscorer onside, only to pull his shot wide off the outside of the post.
Salah strikes twice from the bench to put Stoke to the sword
Former Reds midfielder Allen twice went close in the second half, side-footing a half-volley just wide courtesy of a Dejan Lovren deflection before later blasting over the bar from Crouch's headed knock-down just eight yards out.
And Stoke were made to pay for their failure to force Mignolet into any real action when Mané beat Darren Fletcher to a loose ball, raced to the by-line and clipped a cross into Salah at the back post - the Egyptian unleashing an unstoppable volley past Grant.
But he wasn't done there, and Salah scored his second - and Liverpool's third - as he ran on to Can's over-the-top through ball to dispatch past his 17th of the season in all competitions.
The win lifts Liverpool into fifth, two points above Tottenham Hotspur, with Stoke sinking down to 16th - three points above the relegation zone.
Reds quash game management worries
Liverpool's, or more precisely Jürgen Klopp's, game management had come in for plenty of criticism of late having squandered leads late on in back-to-back games against Sevilla and Chelsea prior to this.
But here Liverpool's second-half display - fuelled by their second-half changes - backed up Klopp's pre-match protestations that claims they cannot manage games have no standing.
Liverpool have scored more goals through substitutes against Stoke than against any other top-flight opponent, with 13 now thanks to Salah's second-half double, and their introductions helped them to take charge at a time when the game was hanging in the balance.
This was the fourth successive game that a Liverpool player has come from the bench to score against the Potters, with Philippe Coutinho and Firmino having both come on to score in a 2-1 win here back in April, added to Divock Origi and Daniel Sturridge's strikes in previous meetings.
And as options off the bench go, there are few better than a free-scoring Salah. So that proved. Liverpool didn't even have to call upon Philippe Coutinho, also rested.
Question marks were raised over Klopp's decision to leave the club's top scorer - Salah - on the bench to start with, but the manager timed the attacker's introduction to perfection and should be praised for doing so.
With Salah on the pitch, Liverpool's attacking threat multiplied and his two goals were a product of his ever-growing confidence.
Stoke could not live with the pace and movement of the Egyptian - undoubtedly the signing of the summer - as Liverpool ran out deserving winners.
There was more than a tad of fortune about their victory, given Liverpool wasted two one-on-one chances in the first 45 minutes and should have had Mignolet sent off.
The rule book states that yellows for a non-deliberate foul are acceptable in the box where a penalty is awarded, avoiding double punishment, but Atkinson's decision let Liverpool off the hook given it was outside of the area.
The away side were also perhaps lucky that Gomez was not adjudged to have let the ball run out of play, a decision which might have gone Stoke's way on another day, before crossing to Solanke for the first goal, though the ball appeared to stay on the pitch by a fraction.
Yet Liverpool defended well and resisted spells of Stoke pressure before hitting them late on to secure an important three points and answer some of the questions asked of their ability to hold out leads in tight games. Rather than scrape victory, this turned out to be a comfortable triumph.
Statistics showed beforehand that Liverpool had conceded more goals beyond the 70th minute in Klopp's 81 league games than any other top-six team, but their substitutions here showed that they too can go on to affect and win matches late on.
Solanke wastes his big chance despite bright display
Handed his first ever top-flight start to add physicality to Liverpool's forward line, it was actually Solanke's fine footwork for Mané's opener where he made his biggest impact.
The 20-year-old saw the least service of the Reds' attacking players in the first half and consistently looked to drop deep to see more of the ball.
But he was excellent for Liverpool's first goal, controlling Gomez's clipped pass with his right before feeding Mané with his left - the Senegalese attacker's close control allowing him to move into space and finish.
But Solanke wasted a gilt-edged chance to mark this occasion with his first league goal on 39 minutes, running clean through on goal after Firmino's flick and firing down the middle at Grant from 20 yards when he had time to venture further forward.
He was not the only player to mis-fire, Mané equally guilty of wasting a good chance before half-time, but it was the kind of opportunity that could determine whether Solanke is afforded similar starting berths in the coming weeks as Klopp looks to use his squad throughout a busy fixture schedule.
Despite that miss - which Klopp pointed out post-match that he should have scored - his performance was full of promise, fine close control and good combination play.
He might even had a second assist, though Mané could not finish from his pass late on in the first period.
And ultimately there was a tinge of regret that he could not add to that a goal to cap it all off.
With Firmino inarguably Klopp's first-choice forward, Sturridge his back-up and even Salah as an option to lead the line - Solanke will be well aware of the miniscule margin for error when he does play.
This will not be his last chance from the start for Liverpool, particularly with so many games to come over the next few months, but he will undoubtedly be disappointed not to have made slightly more of it.
Stoke struggle to threaten
Stoke fans had long been crying out for Crouch's first league start of the season, having scored three goals from the bench to earn five of their 13 league points before this.
But even with the 36-year-old - who signed an extended contract in North Staffordshire on Tuesday - leading the line, their attack was disjointed as they struggled to get players around him to feed off his knock-downs in the first 45 minutes.
They have relied heavily on winger Xherdan Shaqiri this season, his three goals and five assists representing a 50 per-cent direct involvement in Stoke's top-flight goals this season.
And while they threatened in flashes, they desperately lacked in end product. They were notably more involved after Diouf joined Crouch up top and they had a 15-minute first-half spell in which they were the team on top.
Yet Mignolet did not have a save to make, with Shaqiri firing wildly wide and Fletcher's powerful drive from the edge of the area well blocked by Liverpool full-back Gomez.
The hosts attacked with greater vim and vigour for the first 20 minutes of the second half and yet produced nothing from it.
Allen saw a shot deflected just wide before blazing over an excellent chance from eight yards, both from headed knock-downs, before the offside flag spared Diouf huge embarrassment and a potential miss of the season contender late on.
In the end Stoke mustered just one shot on target all game, and that effort only coming in the 91st minute through Jesé Rodriguez.
But Mark Hughes' side simply didn't do enough in the final third of the pitch, despite producing enough moves and openings to do so against a vulnerable Liverpool defence.
Hughes is already under pressure for Stoke's slipping standards over the last 18 months and their rot must not continue.
Having now conceded at least twice in six of their previous seven league games, and with one of the league's worst defensive records across the entire season, flat attacking performances are the last thing Stoke need.