Mohamed Salah scored twice in a 23-minute cameo off the bench to add to Sadio Mane's first-half opener and edge the visitors closer to the Premier League's top four.
Egypt international Salah was one of six players rotated out from the side that drew with Chelsea at the weekend, although the changes failed to prevent the Reds from registering a fourth consecutive away win in this fixture.
Mane, who had not scored in his previous five league appearances, earlier opened the scoring by punishing a sleepy Stoke defence in the 17th minute.
And the Reds all but confirmed another important three points when Salah - who joined the likes of Philippe Coutinho and Jordan Henderson on the bench - fired home from Mane's cross in the 77th minute.
The in-form attacker completed the rout with his 12th league goal late in proceedings as Klopp's side leapfrogged Tottenham, while Stoke - who were aggrieved not to see Simon Mignolet sent off for bringing down Mame Biram Diouf in the first half - slipped to 16th in the table.
Any concerns over the effectiveness of Liverpool's attack without Salah were soon extinguished by Mane's early opener.
They were aided by a static Stoke defence, which paused to protest that Joe Gomez's byline cut-back had run out of play. That enabled Dominic Solanke to turn the ball into the path of Mane, who dinked a clever close-range finish over Lee Grant.
Solanke - in his first league start for the club - should have doubled the advantage when he was freed by Roberto Firmino's back-heel but, with time to pick his spot, the young forward fired straight at Grant.
He almost paid an immediate price as Diouf latched onto a long ball, only to be tripped just outside the area by the onrushing Mignolet.
The goalkeeper escaped with a yellow card despite denying the attacker what would have been a certain goal, and the resulting free-kick came to nothing.
Stoke then received a reprieve of their own on the stroke of half-time, Mane hitting the base of the left upright after being sent through one-on-one by Solanke's pass.
The run of chances continued after the restart as Joe Allen's half-volley deflected narrowly wide off the fit-again Dejan Lovren.
And with Peter Crouch coming more into the match on his first league start of the campaign, Mark Hughes' men began to mount a greater aerial threat.
Crouch's knock-down from an Erik Pieters cross teed up Allen for another promising opening, but the fast-arriving midfielder blazed well over the crossbar.
Klopp responded to the slight loss of control by introducing Salah and James Milner and the former quickly continued his hot run in front of goal, volleying home Mane's cross to make it 2-0 with just over 15 minutes remaining.
And he only needed a further six minutes to double his tally and complete a one-sided victory, outpacing Pieters and applying a cool finish past Grant.
Key Opta stats:
- Liverpool have lost just one of their last 10 Premier League games, winning five and drawing four.
- Liverpool have now made 48 changes to their starting XI this season in the Premier League, 11 more than any other side (Everton next on 37).
- Sadio Mané scored for the first time in six Premier League games for Liverpool, ending his longest top-flight goal-drought with the Reds (5 games).
- Mohamed Salah has had a hand in 14 goals in his 14 games for Liverpool this season in the Premier League (12 goals, 2 assists).
- Salah’s haul of 12 goals in his first 14 Premier League games for Liverpool is a club record.