The first instalment has been repaid with exquisite timing. Five minutes remained at Anfield when Virgil van Dijk towered above the Everton defence and announced his arrival as Liverpool’s record signing with the winning goal in front of the Kop. Some players wilt under pressure and expectation. The most expensive defender in world football soared.

This had been another gruelling derby for Jürgen Klopp against a Sam Allardyce team far more expansive than anticipated. Everton gave their all and were on course for a replay when Gylfi Sigurdsson equalised the second successive soft penalty awarded in an Anfield derby. This time it fell in Liverpool’s favour and was converted by James Milner. An ugly confrontation between Mason Holgate and Roberto Firmino added to the controversy, with the Everton defender understood to have accused the Liverpool striker of using racist language when they clashed in the first half.

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As for the football, this was about vindication for Van Dijk and Klopp. Early days of course, and at £75m Liverpool will expect a greater return than the goal that beat Everton in an FA Cup tie at Anfield for the first time but, as the Holland international put it, this was the dream start. Van Dijk’s aerial command had been to the fore in dealing with Everton pressure and his physical quality paid off in attack when Alex Oxlade-Chamberlain swung over a late corner and he beat Phil Jagielka plus a flapping Jordan Pickford to head in the winner. The first Liverpool player to score on his debut against Everton since 1901 is up and running in style.

Defeat was hard on Allardyce’s team, who produced a much improved and more offensive performance than in the recent 1-1 draw here in the Premier League. Yannick Bolasie’s return from a 12-month injury lay-off was a significant factor with his willingness to attack the impressive Andrew Robertson offering Dominic Calvert-Lewin necessary support. Everton, however, did not create one chance of note in the first half. That will not have shocked regular observers of this team, who have sought to rectify their season’s biggest failing by signing Cenk Tosun for £27m from Besiktas. The club bizarrely announced the completion of the deal at half-time.

More surprising was Liverpool’s equal ineffectiveness in the final third. In mitigation Klopp’s forward options were reduced or lacking sharpness. Sadio Mané arrived back from the African Footballer of the Year awards in Ghana only in the early hours of the morning via private jet. The winner, his team-mate Mohamed Salah, was absent through injury along with the transfer window distraction that is Philippe Coutinho. Salah’s replacement, Adam Lallana, was on the periphery of a contest played in a good atmosphere until being central to the breakthrough.

The only incident of note in the opening 25 minutes was a poor challenge by Wayne Rooney on Joe Gomez that brought a deserved booking. Otherwise it had been a flowing, even but incident-lacking derby until Emre Can found Lallana inside the penalty area and the boyhood Evertonian went down under a touch from Holgate. The referee, Robert Madley, immediately pointed to the spot and the controversy that shaped the last derby, and earned Everton a point, had resurfaced. Holgate raised his arms foolishly, Lallana tumbled easily, and a highly contentious penalty resulted. Milner dinked his spot-kick straight down the middle of Pickford’s goal but showed barely a flicker of emotion at scoring.

Facebook Twitter Pinterest The referee Robert Madley separates Mason Holgate and Roberto Firmino as the two players square up. Photograph: Carl Recine/Action Images via Reuters

Allardyce argued over the decision with Jon Moss, the fourth official. Given their respective takes on Dejan Lovren’s push on Calvert-Lewin last month, the Everton manager would have agreed and Klopp disagreed with Madley’s judgement if they were being consistent. In fairness to the Liverpool manager, he was. “For me it was exactly the same situation as the last game,” he said. “You all said the hands were there and it’s a penalty. I don’t think both situations were a penalty.”

Another flashpoint involving Holgate soon followed. The young defender stupidly pushed Firmino over the advertising hoardings in front of the Main Stand as the pair chased the ball out for a throw-in. Firmino reacted furiously and made a bee-line for Holgate. Madley stepped in the Liverpool forward’s path as the Everton defender first eyeballed Firmino then erupted in response to something the Brazilian said, man-handling the referee in the process. Remarkably no cards were shown but Madley engaged in a long conversation afterwards with Moss, who took notes. The match official’s report will shape what action, if any, the Football Association takes next.

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A dubious penalty was not the only similarity with the clubs’ recent league meeting. Again, Liverpool should have put the contest beyond Everton at 1-0. Joe Gomez squandered a gilt-edged chance early in the second half when Alex Oxlade-Chamberlain’s corner found him completely unmarked at the back post. Lallana also shot wide when given a run on Phil Jagielka and van Dijk headed straight at Pickford from an Oxlade-Chamberlain free-kick.

Again, just like in the league meeting, Everton capitalised. Allardyce introduced Ademola Lookman for the unhappy Rooney, with the former England captain sailing too close to a red card, and the youngster was central to the equaliser. Collecting Bolasie’s clearance, he broke into the Liverpool half before spraying the ball out to Jagielka of all people sprinting down the right. Jagielka stumbled over his first touch but passed inside to Sigurdsson and he rolled a cool finish inside Loris Karius’ left-hand post.

Klopp lamented the lack of protection given for Everton’s equaliser. He gave thanks, by contrast, for changing his mind and deciding to play Van Dijk after all.