Sadio Mane scored once and Mohamed Salah stepped off the bench to bag a brace and take his tally for the season to 17 goals.
But Stoke ended the game with a seething sense of injustice.
They were unhappy that Liverpool’s opening goal was allowed to stand and furious that visiting goalkeeper Simon Mignolet was not sent off just before half-time.
Klopp’s men, though, made maximum use of whatever good fortune came their way with enough clinical moments to kill off their hosts.
And, after they squandered leads in their last two games against Sevilla and Chelsea, they rarely looked likely to be so generous to Mark Hughes’s side, whose shoddy defending contributed to a depressing night.
If supporters braved the icy Potteries climate early in the hope of an exciting start to the game, they were disappointed.
A series of niggling fouls, overhit passes and poor decisions punctuated the first 15 minutes with neither goalkeeper called upon to exert himself.
A comfortable catch from Stoke’s Lee Grant to claim a cross by Alex Oxlade-Chamberlain was the most taxing moment for either gloveman.
So neutrals were grateful for the controversy that delivered the opening goal on 17 minutes, even if Stoke were less than appreciative.
Joe Gomez raced forward from right-back for Liverpool to chase a ball into the corner but Stoke were certain the ball had crossed the line before the defender crossed.
Replays were inconclusive and the officials were unmoved, allowing Dominic Solanke to help the cross deftly into the path of Mane, whose chipped finish was too good for Lee Grant in the Stoke goal and crept over the line.
Grant made a comfortable save from Alex Oxlade-Chamberlain before Stoke finally gained a foothold in the game with Xherdan Shaqiri at the heart of their best play.
He was not involved when Darren Fletcher had a shot blocked after an acrobatic assist from Mame Diouf, but the Switzerland star’s movement and quick feet caused problems for Liverpool, not least when they allowed him space for a cross that Joel Matip had to clear.
Shaqiri was then involved in releasing Joe Allen for a cross that Peter Crouch headed just wide before Grant denied Solanke after a clever exchange of passes with Roberto Firmino.
It was then that Stoke’s sense of injustice intensified with a contentious decision by referee Martin Atkinson.
The official awarded a free-kick for Mignolet’s challenge brought down Diouf just outside the box, but only showed a yellow card when Diouf had appeared certain to score.
The free-kick came to nothing, leaving Stoke furious that the Reds had escaped with their lead intact and their team at full-strength.
Stoke’s fans were livid and manager Mark Hughes appeared extremely unimpressed with most observers believing Klopp’s men had received a major let-off.
Mane ought to have rubbed salt into the wound when Stoke failed dismally to catch him offside and he raced clear, only to strike the base of the woodwork with only Grant to beat.
That was the final action of a first half that had started slowly but come to life, and Atkinson had to handle the wrath of angry home supporters as he blew the half-time whistle.
Stoke ought to have been level just three minutes into the second half when Shaqiri caused more problems with an inswinging cross that Liverpool struggled to deal with.
Matip cleared it partially under pressure from Crouch but it dropped invitingly for Allen, whose low shot struck a grateful Lovren and skimmed behind for a corner, which the visitors cleared.
But the hosts struggled to follow up that near-miss as Liverpool’s composed play kept Hughes’s men at arm’s length from Mignolet’s goal.
Stoke finally had some joy when the visitors failed to spot Allen’s run from midfield to the byline, with the Welshman forcing a corner. But the flag-kick was cleared by the Reds who caused problems on the break, with Stoke forced to clear hurriedly away from their six-yard box after a pass from Alberto Moreno.
The Potters claimed another free-kick on the left but the inswinging delivery from Darren Fletcher was punched clear by Mignolet.
There was more anger with 23 minutes remaining when Stoke defender Bruno Martins Indi produced a crude block with his arm on Oxlade-Chamberlain, who reacted by shoving the Dutchman in the chest. Both men were booked and, in the aftermath, Stoke should have drawn level.
A cross by Erik Pieters was headed down by Crouch for Allen, who blasted over.
With 13 minutes left Liverpool ended any doubts about the win as Mane crossed and Salah volleyed home. Six minutes later Salah added the third when he beat Pieters to a diagonal ball, raced clear and fired low past Grant.
Grant; Zouma, Shawcross, Martins Indi; Diouf, Fletcher (Adam 85), Allen, Pieters; Shaqiri, Crouch, Choupo-Moting (Jese 75).
Mignolet; Gomez, Matip, Lovren, Moreno; Oxlade-Chamberlain (Milner 67), Can, Wijnaldum, Mane (Henderson 89); Solanke (Salah 67), Firmino.