Watch: The last time Liverpool & Everton met in the FA Cup in 2018

Virgil van Dijk made a dream start for Liverpool following his £75m move from Southampton as his late header settled a highly competitive FA Cup third round Merseyside derby against Everton at Anfield.

The visitors looked on course for a replay at Goodison Park until keeper Jordan Pickford misjudged a corner, allowing Van Dijk, who cost a world record fee for a defender, to steal in six minutes from time and send the Reds into the fourth round.

Gylfi Sigurdsson's smooth finish from the edge of the area put Everton level in the second half after Liverpool had taken the lead 10 minutes before half-time, with James Milner scoring from the penalty spot following Mason Holgate's tangle with Adam Lallana.

In a fiercely fought game there was a first-half flashpoint when Holgate foolishly pushed Reds forward Roberto Firmino over the advertising hoardings and into the crowd, with the Everton defender left furious after the pair subsequently exchanged words.

Liverpool were without injured forward Mohamed Salah and midfielder Philippe Coutinho, who has been linked with a £140m move to Barcelona.

Van Dijk makes the dream start

Liverpool manager Jurgen Klopp wasted no time in introducing Van Dijk to his defence after he finally made his move from Southampton - but it was in attack where the 26-year-old Netherlands international fulfilled his dreams in front of the Kop.

He had already wasted one chance to mark his debut with a goal when he could only head at Pickford from six yards. He made no mistake when presented with that crucial opportunity six minutes from time.

Van Dijk got to the ball ahead of Pickford, whose rash attempt to claim was heavily punished as Anfield went into ecstasy.

It is in defence where he will be expected to do his main work, but this was a highly impressive debut at both ends for the Dutchman.

Van Dijk cuts an imposing figure and was almost unbeatable in the air up against the game and tireless Dominic Calvert-Lewin.

Greater tests than this lie ahead. Everton boss Sam Allardyce was unable to use the threat he hopes will come with the £27m arrival of Turkey striker Cenk Tosun, so Everton's danger was limited.

As a starter, though, this was exactly what Klopp and Liverpool had in mind when they handed over £75m to Saints.

And to make his debut even better, he was named TV man of the match by the BBC experts at the game.

Busy night for ref Madley

Merseyside derbies can always prove difficult occasions for referees - and this was no different as Bobby Madley had to deal with several contentious incidents.

Madley booked Wayne Rooney as early as the seventh minute for a reckless challenge on Joe Gomez - and he needed to keep an eye on the boyhood Evertonian as he sailed very close to the wind, and a second yellow card, before being substituted after 52 minutes.

He was in the action again on 35 minutes when he pointed to the spot after Holgate tangled with Lallana. Everton will complain it was a soft award - as Liverpool did when Dejan Lovren was punished with a penalty when he manhandled Calvert-Lewin in the recent 1-1 draw at Anfield - but once the Toffees defender laid hands on his opponent he was asking for trouble.

The biggest flashpoint came moments later when Holgate had a rush of blood to the head and shoved Firmino into the crowd. The incident boiled over as the Brazilian rushed back at Holgate to protest - angry words were exchanged and Everton's youngster reacted with undisguised fury to something Firmino said.

After that, Madley was able to keep a check on the emotions as a game that was passionate never threatened to get out of hand.

Can Tosun solve Everton's problems?

With interesting timing, Everton announced the completion of the £27m deal for Besiktas striker Tosun during the half-time interval.

And while their display was a vast improvement on the attritional effort in the recent draw at Anfield, it is clear the Toffees need the sort of spearhead Allardyce hopes Tosun will provide.

The visitors at least carried some threat here via the pace and width of Yannick Bolasie and substitute Ademola Lookman, but they require the sort of finishing power that has been missing with the failure to replace Romelu Lukaku.

And one final note on Everton: keeper Pickford will take responsibility for the winner but it should also be remembered he has been magnificent so far in what has been a very difficult season at Goodison Park.

More Merseyside derby success for Liverpool - the stats

Liverpool are now unbeaten in 16 matches against Everton (won eight, drew eight) - their longest ever unbeaten run against them.

Everton were unbeaten in their first seven matches under Allardyce (won four, drew three) but they have lost their past three, scoring just two goals in their past five matches.

Milner has scored 11 of his 13 penalties for Liverpool - though he had failed to score with his previous two efforts before Friday.

Milner has scored three goals in his past two FA Cup third-round matches, also scoring twice for Manchester City against Sheffield Wednesday in January 2015.

All four of Sigurdsson's goals against Liverpool have been scored at Anfield, with two in the FA Cup.

The Reds have not lost a Merseyside derby in which they have scored first since December 1992.

This was the 24th meeting between these sides in the FA Cup, making it the most-played fixture in the competition's history.

'Not a penalty' - what they said

Liverpool manager Jurgen Klopp to BBC Sport: "It was not the best piece of football for 95 minutes, but a proper fight. I'm delighted with the attitude we showed again - it was a difficult game. It was not the night for brilliant football.

"You would love to see that but Everton defended well. They played football, a big difference to the last game between us.

"The equaliser was the worst protection I ever saw in my life. It made no sense at all. We were all in the wrong direction. But when you make mistakes in football you have to react and we did again.

"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.

"It was a similar situation [to the penalty against Liverpool for Everton last month]: the first one was no penalty in my opinion and this one wasn't also. It's the same.

"It was not necessary for Mason Holgate to give Firmino that push. It was really dangerous and usually players don't do that."

Liverpool 2-1 Everton: Jurgen Klopp admits both derby penalties should not have been given

Everton manager Sam Allardyce to BBC Sport: "We gave everything we could today and in the end we were disappointed we did not take them back to our place. It is a shame after all that effort.

"I didn't think it was a penalty, but there is not much we can do now. For me, I thought some of the fouls they got and bookings we got and fouls we didn't get was very harsh by the referee in terms of consistency. There were similar incidents, similar penalties.

"At the end of the day, we have given everything we could and we tried our very best today."

Asked about the incident between Holgate and Firmino, Allardyce replied: "I'm only interested in the football, not want happened with Firmino and Mason. I will definitely ask him and we will see where it goes from there."

What's next?

Both teams are next in Premier League action, with Everton away at Tottenham on Saturday, 13 January (17:30 GMT kick-off), before Liverpool host leaders Manchester City at 16:00 the following day.