The race to the finish line has started already according to Jürgen Klopp and by those punishing standards Liverpool have faltered badly. Swansea City belied their league position, their form and their history here to rise off the foot of the Premier League with an accomplished display that reverberated all the way to the top.

A cold, flat and goalless first half gave way to a remarkable second when Fernando Llorente gave Paul Clement’s team a two-goal lead only for Roberto Firmino to lead a comeback and level matters. Liverpool sensed another statement victory in their pursuit of the title but another defensive lapse enabled Gylfi Sigurdsson to seal Swansea’s first league win here. A first home league defeat for more than a year left Klopp and his players feeling bad and the hangover will be even worse should Chelsea open up a 10-point gap with victory over Hull City on Sunday.

Klopp claimed Clement had instigated the fastest improvement in a relegation-threatened team he had ever seen despite last Sunday’s 4-0 reverse against Arsenal. It proved to be no exaggeration. “The mindset in the group is that anything is possible,” said his Swansea counterpart.

“Our main objective is not to look at the table or talk about how many wins or points we need but to just focus on delivering some performances. Against Arsenal we delivered a 35-minute performance. We suffered two big blows that put us on the back foot, two own goals that were very unlucky, but today we gave a 90-minute performance. We defended well, we took our opportunities at set-pieces and were dangerous on the counterattack.”

Klopp lamented Liverpool’s failure to take a first-half lead but not the merits of Swansea’s victory. Maybe it was the early kick-off or the freezing temperatures but the hosts were subdued until falling two goals behind. Philippe Coutinho initially offered his customary range of inspired passing on his first home appearance since damaging ankle ligaments in November but even he was running the ball out of play by the end of an unusually slow, cumbersome opening by Liverpool’s standards.

Emre Can and Adam Lallana squandered decent openings before the break while Swansea almost profited from their first attack, with new signing Tom Carroll striking a post via a deflection off Dejan Lovren. Carroll and Martin Olsson impressed on their Swansea debuts having arrived at the club in midweek.

Liverpool came out early for the second half but did not switch on. Lovren needlessly conceded a corner with a miscued header and from Sigurdsson’s delivery Liverpool lost every subsequent challenge. Federico Fernández beat Ragnar Klavan and Lovren in the air to head goalwards. The ball rolled off Llorente’s shins on to Wayne Routledge with his back to goal and after he controlled the centre-forward stabbed the loose ball beyond Simon Mignolet from close range.

“Llorente is completely alone in our six-yard box,” said Klopp. “It should not be difficult to miss him.”

His side were two goals in arrears before they could rouse themselves. Swansea’s new signings were at the heart of an outstanding strike as Olsson broke and released Carroll down the left. The midfielder swept a perfect first-time cross towards the penalty spot and there was Llorente, towering above three defenders, to steer a textbook header beyond Mignolet’s grasp. Swansea were in dreamland, only now the hosts awoke from their slumbers.

The visitors had the cushion of a two-goal lead for three minutes before Firmino converted a convincing header of his own. Jordan Henderson sprayed a cross-field ball to James Milner on the left and he returned a deep cross towards the Brazilian who, with Olsson failing to clear, beat Lukasz Fabianski with an unstoppable header. Olsson claimed he was nudged in the back by Firmino and may have had a point, but the referee, Kevin Friend, was unmoved.

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Liverpool, team and crowd, were finally stirred and they clawed back the deficit courtesy of a sublime finish from the Brazil international. Georginio Wijnaldum created the equaliser with a fine piece of control to kill Milner’s high ball and cross from the left but it was all about Firmino. Two defenders were in close attention yet powerless to prevent the forward taking the ball on his chest and sweeping a first-time finish inside Fabianski’s right post.

The stage was set for the home side to complete the fightback. Time, momentum and Swansea’s understandable nervousness was on Liverpool’s side but so, too, was weak defending and it was the visitors who struck for a third time. Leroy Fer and Llorente combined for Carroll to attack the Liverpool rearguard. He evaded two nondescript challenges and over-ran the ball into the path of Klavan, who could only divert it across the penalty area towards Sigurdsson. The Iceland international made no mistake as he sent the ball back across Mignolet’s goal.

Liverpool fought back, Lallana striking the bar and wasting the rebound, but to no avail.

Klopp insisted damaging was too harsh a word to describe the impact of this result on his team’s title chances. Harsh but fair.