Match ends, Stoke City 0, Liverpool 3.
Mohamed Salah came off the bench to score twice and secure an ultimately comfortable away victory for Liverpool at Stoke that moves Jurgen Klopp's side up to fifth in the Premier League.
Senegal winger Sadio Mane gave the Reds a controversial early lead, dinking the ball into the net after a marginal linesman's call.
Stoke believed Liverpool keeper Simon Mignolet should have been sent off for a challenge just outside the box on their striker Mame Biram Diouf.
Mark Hughes' side applied more pressure after the break but Egypt forward Salah gave his side a two-goal cushion with a searing volley from Mane's cross.
The in-form forward then sealed victory by scoring the third with a composed finish when he was gifted a clear run on goal.
Liverpool overtake Spurs into fifth and are two points behind fourth-placed Arsenal, while Stoke - with just one win in their last seven league games - are three points above the drop zone at the other end.
Salah delivers value and goals
A study by the CIES Football Observatory in September rated Salah as the most under-valued purchase in the summer transfer market, estimating he was worth about double the initial £34m Liverpool paid Roma in July.
For once, the algorithms have been borne out on the pitch.
The Egyptian reclaimed sole possession at the top of the Premier League scoring charts with his 11th and 12th league strikes of the season, goals that have arrived at a rate of one every 87 minutes.
Unleashing his pace and freshness off the bench in the 67th minute demonstrated the increased strength in depth of Klopp's squad this season.
Salah duly made a swift and decisive impact, thumping home Mane's cross - after the former Southampton man had raced past Darren Fletcher on the right - before latching on to a weak Erik Pieters back pass and slotting home his side's third.
Klopp's starting line-up - with his biggest goal threat left on the bench, along with Philippe Coutinho - was brave, but the German was vindicated by his side achieving a good result at a difficult venue while giving Salah a rest with the festive period looming.
Atkinson bears brunt of Stoke frustration
While Salah was the key figure by the end, for much of the game it had been referee Martin Atkinson.
The official left the pitch at the end of the first half surrounded by security staff and the focus of camera lenses and jeers from the home crowd.
His interpretation of a string of midfield tussles and his dismissal of a Peter Crouch penalty appeal irritated the Stoke fans - but it was two key decisions that had escalated the mood to rage.
Firstly, Joe Gomez's cross, which led to Liverpool's opener, was delivered with most, if not all, of the ball having already crossed the byline.
With repeated replays unable to show whether the ball was out, it was an almost impossible call in real time.
Then Mignolet lunged recklessly outside his penalty area and sent Diouf tumbling six minutes before the break.
Stoke's Twitter account was certain it should have been a red card, given the Senegal striker was heading round the Belgium goalkeeper towards a goal-scoring opportunity.
"That's the key moment in the game from our point of view," said Hughes.
"I just don't understand why the referee and the officials couldn't see that for what it was - a clear goal-scoring opportunity."
Klopp took a different view. "There were two covering players in our shirt. That's the reason, that's the rule," he said.
Hughes under the microscope
Klopp's side surrendered leads against Sevilla and Chelsea in their last two matches and Stoke gave the German a second half to mostly endure rather than enjoy.
With Eric Maxim Choupo-Moting and Diouf joining Crouch in a sky-scraping forwardline, the visitors' suspect defence often creaked under an aerial attack.
Liverpool kept their structure and their heads, though, to claim a first away clean sheet of the league season.
The Reds were grateful for old boy Joe Allen's profligacy, however, as the Wales midfielder twice dragged shots off target from prime positions in the penalty area after the break.
Hughes will point to those fine margins and the first-half's tight calls, but the direct style that has been successful in recent seasons now has the Potters perched perilously above the drop zone.
With just two points from their last four games and back-to-back defeats, Hughes could find he is the next manager under the microscope if the Potters' nine-season stay in the top flight becomes seriously threatened.
Man of the match - Joe Gomez
'They've got away with one' - analysis
Former Liverpool midfielder Danny Murphy on Match of the Day: "It should be a sending off for Mignolet. Diouf is ahead of Joe Gomez and he would have put the ball in an empty net. They've got away with one."
Former Arsenal and England striker Ian Wright on Match of the Day: "It's absolutely a sending off for Mignolet. It's a sending off.
"I am worried about Stoke, they are not getting any luck. I am concerned for Mark Hughes, it's very precarious for managers at the moment."
Liverpool rotate to find winning combination - match stats
- Liverpool have now made 48 changes to their starting XI this season in the Premier League, 11 more than Everton - the next most-changed side.
- Liverpool have lost just one of their last 10 Premier League games, winning five and drawing four.
- Mane scored for first time in six Premier League games for Liverpool, ending a five-game top-flight goal drought - the longest he has endured for the Reds.
- Salah has had a hand in 14 goals in his 14 games for Liverpool in the Premier League (12 goals, 2 assists).
Stoke meet Swansea at home at 15:00 GMT on Saturday, with Liverpool travelling to Brighton at the same time.
- 15Martins IndiBooked at 65mins
- 24FletcherSubstituted forAdamat 85'minutes
- 10Choupo-MotingSubstituted forRodríguez Ruizat 74'minutes
- 11Rodríguez Ruiz
- 22MignoletBooked at 39mins
- 23CanBooked at 56mins
- 5WijnaldumBooked at 63mins
- 21Oxlade-ChamberlainBooked at 66minsSubstituted forMilnerat 67'minutes
- 19ManéBooked at 74minsSubstituted forHendersonat 89'minutes
- 29SolankeSubstituted forSalahat 67'minutes
- Martin Atkinson
- Shots on Target
Second Half ends, Stoke City 0, Liverpool 3.
Corner, Liverpool. Conceded by Joe Allen.
Corner, Liverpool. Conceded by Bruno Martins Indi.
Offside, Stoke City. Jesé tries a through ball, but Mame Biram Diouf is caught offside.
Attempt saved. Jesé (Stoke City) right footed shot from outside the box is saved in the centre of the goal.
Foul by Jordan Henderson (Liverpool).
Xherdan Shaqiri (Stoke City) wins a free kick in the defensive half.
Attempt blocked. Roberto Firmino (Liverpool) right footed shot from outside the box is blocked.
Substitution, Liverpool. Jordan Henderson replaces Sadio Mané.
Corner, Liverpool. Conceded by Lee Grant.
Attempt saved. Georginio Wijnaldum (Liverpool) right footed shot from outside the box is saved in the top centre of the goal.
Sadio Mané (Liverpool) wins a free kick in the attacking half.
Foul by Charlie Adam (Stoke City).
Attempt saved. Roberto Firmino (Liverpool) right footed shot from outside the box is saved in the centre of the goal. Assisted by Alberto Moreno.
Substitution, Stoke City. Charlie Adam replaces Darren Fletcher.
Offside, Stoke City. Ryan Shawcross tries a through ball, but Mame Biram Diouf is caught offside.
Goal! Stoke City 0, Liverpool 3. Mohamed Salah (Liverpool) left footed shot from the centre of the box to the bottom left corner following a fast break.
Attempt missed. Sadio Mané (Liverpool) header from the centre of the box is close, but misses the top right corner. Assisted by Mohamed Salah with a cross following a corner.
Corner, Liverpool. Conceded by Erik Pieters.