It looked like the replay was on, a trip over the road to play another leg-sapping Cup tie against Big Sam’s side.
Instead, the £75m got it right on the night.
This win extends Liverpool’s unbeaten run against Everton to 16 games, a statistic that nobody involved with the Blue Noses can tolerate.
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The Brazilian was sent tumbling into the crowdCredit: PA:Press Association
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Firmino raced back onto the pitch to confront Holgate and unleashed a verbal attack on the Blues starCredit: Reuters
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Players from both sides were forced to intervene to split up the pairCredit: Carl Recine
They will have to do something about that in the years to come under Big Sam.
Nobody, with the exception of the manic figure of Everton old-boy Peter Reid in the studio, really believed they could pull it off last night.
Liverpool had gone ahead after 35 minutes, stroking his penalty beyond Jordan Pickford with a simple, classy finish. Easy.
Adam Lallana had been quiet, feeling his way back into this Liverpool side after a lengthy spell out through injury.
For many reasons, not just this one, it was not Mason Holgate’s night out there in a blue shirt.
Naively he wrapped his arms around Lallana, rolling him to the floor WWF-style in front of referee Bobby Madley.
Milner, captaining the Liverpool side in the absence of Jordan Henderson, made no mistake.
Pickford came flying out, leaving his line early to try to meet Milner’s penalty, but he was already wrong-footed.
So was Firmino when Holgate shoved the Liverpool flyer over the advertising hoardings and into the home fans.
What happened next takes some unravelling.
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Wayne Rooney received a first half caution at Anfield as he struggled to make an impactCredit: Reuters
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He was replaced by Ademola Lookman shortly after the breakCredit: Reuters
Firmino, understandably, reacted badly after he bounced back out of the crowd and on to the pitch.
Madley stepped in to separate them, but Everton’s young defender lost his rag when Firmino appeared to spit out an insult in his direction.
This third round tie, full of passion and intensity, was coming nicely to the boil.
Wayne Rooney had made this mark early doors with a nasty challenge, a reducer on Liverpool’s central defender Joe Gomez.
It was a vicious challenge, a straightforward booking for the Everton forward. After that, he was walking a tight-rope.
When he tangled with Emre Can at the start of the second half, the former England captain was one more indiscretion away from being sent down the tunnel.
Allardyce, making five changes to the side beaten by Manchester United on New Year’s Day, needed a shaft of light.
A Rooney through-ball, a Rooney moment, a Rooney belter from the edge of the box.
Instead Big Sam sensibly took him, substituting his hot-head forward after 52 minutes to save him the bother of a sending off.
Everton, struggling for form in the Premier League again, do not need Rooney serving a three match ban.
FORMER PREM REF MARK HALSEY
LIVERPOOL received a soft penalty in the derby last month — and this one last night was even worse.
Adam Lallana went down after minimal contact from Everton defender Mason Holgate in the area.
Was he deceiving the referee? I’d say, yes.
Holgate was not holding the forward and Lallana has tried to deceive the ref to gain an advantage. Referee Bobby Madley is low on confidence and I’m surprised he was given this calibre of game given his recent performances.
The FA Cup is important, but Big Sam will sacrifice a little bit of glory along the way to save their skins in the Premier League.
Everton equalised, with Yannick Bolasie, substitute Ademola Lookman and Phil Jagielka of all people involved in the build-up to the equaliser.
Sigurdsson, a talismanic figure in the colours of Swansea, is starting to get a few important goals in an Everton shirt. This was one of them.
He scored in front of Everton’s fans, convincing them that they would go head-to-head in a replay at Goodison Park. It was on.
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Adam Lallana won the penalty for Liverpool in front of the travelling Everton fansCredit: Reuters
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Gylfi Sigurdsson finished off a fine move after being teed-up by Everton team-mate Phil JagielkaCredit: Reuters
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The Iceland international is mobbed by his Everton colleagues after bagging the equaliserCredit: AFP
Everton grew in confidence, with Allardyce urging his players to pour forward in search of a famous Anfield winner. The players wanted as well.
That was until Liverpool won the corner that led to Van Dijk scoring the winner with five minutes left on the clock.
He had looked classy and composed at the back, given the run of the place until Sigurdsson scored for Everton.
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Virgil van Dijk could not have wished for a better Liverpool debutCredit: PA:Press Association
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The Dutch defender headed home a corner to spark jubilant scenes at AnfieldCredit: Reuters
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Everton's stars look on in disbelief as Van Dijk raced away to celebrate his dream debutCredit: Reuters
That prompted Liverpool to react, adding a little bit of quality to their attacking moves in the final 10 minutes.
Van Dijk responded, meeting the inswinging corner to place his header beyond Everton’s keeper.
It was a classy finish, the sort of goal that will give him confidence after his first win in a red shirt.
After this, he deserves to wear it.
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Jurgen Klopp was delighted with his £75m new boy's impactCredit: AFP
WHAT THEY SAID
Liverpool hero Virgil van Dijk told BBC Sport: "It felt amazing to be out there and the goal was very special for me and my family. It’s a dream come true."
Kop boss Jurgen Klopp added: "It was nice that Virgil van Dijk could score in his first game, a brilliant header. I am really happy.
"In front of the Kop, makes it even better. In Germany we say cup games have different laws and it was like that - there was an extra push form both sides."