Spanish international Fernando Llorente was the hero in Swansea’s first league victory under new manager Paul Clement, toe-poking the ball home two minutes after halftime and heading home another five minutes later.
After Roberto Firmino had dragged Liverpool back into the game with two superb goals, it seemed wholly unlikely that the visitors could survive the onslaught until the ball broke for Gylfi Sigurdsson to slot home their 74th minute winner.
This was the first of four games at Anfield in three separate competitions over the rest of the month that will do much to shape Liverpool’s season and both Klopp and his bench looked crestfallen.
“We came back and played really good and the most disappointing moment was the third goal. I have no explanation for this,” Klopp told BT Sport, feeling that Sigurdsson should never have been allowed the freedom of the box.
Liverpool, who welcomed back Philippe Coutinho for his first league start since November, dominated proceedings from the opening minutes but found Clement’s side, including new signings Tom Carroll and Martin Olsson, compact and organised.
Emre Can should have done better with a free header from Jordan Henderson’s pinpoint delivery and Adam Lallana ballooned an extravagant mid-air volley high.
Yet for all of Liverpool’s domination of possession, these were rare openings and the nearest to a first-half goal was when Carroll, signed from Tottenham Hotspur in midweek, saw his shot from a narrow angle deflect off Dejan Lovren and strike the post.
Still, there was no sign of the drama immediately after halftime when Llorente struck twice, his second after 52 minutes being a magnificent, soaring header from Carroll’s cross.
Liverpool roared back, Firmino heading in James Milner’s 55th minute cross and then producing sublime skill to take Georginio Wijnaldum’s looped cross on his chest before half-volleying home a thunderous left-foot strike.
With Anfield baying for a comeback winner, it seemed just a matter of time but when Carroll made a rare foray towards the heart of Liverpool’s defence, the ball fell kindly for Sigurdsson and the Icelander was typically cool with his precision finish.
Liverpool came desperately close to equalising when Lallana’s deflected shot hit the bar and his follow-up header flew over but this was a dreadful setback for a team that has lost momentum, having won just one of its last six matches in all competitions.
(Reporting by Ian Chadband; Editing by Brian Homewood and Pritha Sarkar)