By the end Southampton’s support seemed unsure as to where they should direct their scorn. They had booed Virgil van Dijk throughout his return to the south coast, merely grumbling at Sadio Mané’s incisive contributions, while Adam Lallana and Dejan Lovren were heckled during their own late cameos. Yet, as plenty of the home fans flooded forlornly away, it was Mauricio Pellegrino who shivered through the last rites of this latest defeat. The chorus of discontent from those who remained at the final whistle told its own story.

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There can be no living with Liverpool on their travels at times, so incisive is their play and slick their passing. Teams near the foot would not normally consider a visit from Jürgen Klopp’s progressive side as an opportunity to instigate a revival. But Southampton’s was still a meek surrender. Swansea had demonstrated last month what can be achieved against this opposition with a blend of resilience and ruthlessness, and yet Pellegrino’s team offered neither. Sloppiness presented their visitors with an early lead and their second-half display amounted to little more than a whimper.

That lack of response, even against superior opponents, reflected the fragility of the mood on a day when eye-catching wins by Huddersfield and Newcastle had cast this team into the relegation places. All the positivity generated by that victory at West Bromwich Albion has dissipated. That remains their solitary league success since November which puts talk of a revival – they had actually been unbeaten in six matches in all competitions prior to this game – into proper context. “We had to be braver,” said Pellegrino. “The biggest ‘defeat’ was the way we played the second half, for me. Little by little our confidence dropped.”

A glance at the home side’s bench, where there is quality and experience aplenty, might normally offer encouragement that a recovery can be mounted over the last 11 games. But in Southampton’s case it tends to expose under-achievement. They may be one victory from 13th in a congested division but, as a squad, they are surely capable of far better than this. They are a team blunted by Charlie Austin’s injury and, dare it be said, left exposed by Van Dijk’s defection. “Liverpool didn’t have to do much to win easily,” Pellegrino said. “One mistake, one amazing one-two, and that finished the game.”

That was brutally honest. His team were dismissed here by a side gearing up for the resumption of the Champions League and now persuasively targeting a top-two finish. Liverpool have scored 34 away goals this season, more than 11 top-flight teams have managed in total, with the energy and precision of Mané, Mohamed Salah and Roberto Firmino utterly irresistible. This team are equally adept at flooding forward on the counter-attack or pinning opponents so deep that they eventually crack. Their first-half goals reflected both skills, even if each benefited from jittery Southampton defending.

Wesley Hoedt endured life as the fall guy. He and Mario Lemina had already almost contrived to liberate Firmino beyond their own back-line when, six minutes in, Loris Karius bowled the ball out to Alex Oxlade-Chamberlain on the right flank for the midfielder to sprint and bend a pass upfield towards Salah. The delivery was ambitious but Hoedt fluffed his attempted interception and duly presented the Egyptian with space in which to charge. Cedric Soares tore across in a vain attempt to intercept, but Salah kept his head and merely squared for Firmino, one of two team-mates who had arrived untracked at his side, to convert.

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There had been barely 10 seconds between Van Dijk nodding the ball back to Karius and Alex McCarthy being breached at the other end. The boos marking the Dutchman’s touch were choked and drowned amid the raucous celebrations of those in the away section. “It’s not too complicated,” Klopp said of the ferocity of that counter-attack. “The transition at the start is most important, so that first sprint from Ox, that was very important. Then the pass, that little miss ... it was just good.” It deflated the mood in an instant.

Briefly, tantalisingly, Southampton had threatened an equaliser with Karius doing well to deny Pierre-Emile Hojbjerg and James Ward-Prowse. Yet in what time remained before the break Liverpool merely responded with a spell of pressure, squeezing the conviction from their hosts as they pressed them deeper into retreat. Mané might have flung over a centre in hope from the left but chose instead to retain possession and find his deepest-lying team-mate, Joël Matip. The centre-half, still well inside Southampton territory, slid the ball to Salah on the edge of the area and, under only vague pressure from Oriol Romeu, the Egyptian exchanged passes with Firmino. The Brazilian’s back-heel beyond Jack Stephenswas perfectly placed and paced. Salah collected and converted crisply into the corner for a 29th goal of term.

Everything thereafter was a procession, Liverpool passing up opportunities to add a third but the game long since won. They will venture into the Champions League this week confident Porto can be wounded. For Southampton next weekend’s FA Cup tie is a welcome distraction in a troubled campaign.