New Liverpool signing Virgil van Dijk made a perfect start to life at the club by scoring an 84th-minute winner to hand his side a 2-1 victory over Merseyside rivals Everton in the third round of the FA Cup this evening.

The Reds had taken the lead through a first-half James Milner penalty but, just as they did in the Premier League meeting between the two sides last month, Everton looked to have rescued a draw with Gylfi Sigurdsson's second-half equaliser.

However, it was left to Liverpool's £75m man - the most expensive defender of all time - to write the headlines on his debut when he planted his header home with just six minutes remaining to fire Jurgen Klopp's side into the fourth round.

Van Dijk's inclusion was one of six changes made by Klopp from the New Year's Day victory over Burnley, while Everton made five changes of their own from the defeat to Manchester United on Monday.

The changes seemed to affect the flow of both sides as the match got off to a slow start, with an early Wayne Rooney yellow card for a heavy challenge on Joe Gomez and a Sadio Mane long-range strike which flew well off target the only moments of any note in the opening 25 minutes.

The first semblance of a chance fell to the home side when Gomez picked out an unmarked Milner on the left-side of the penalty area, but his first-time volley was just too far in front of Adam Lallana on its way past the far post.

It wasn't until 10 minutes before half time that either side even registered a shot on target, but it provided the opening goal as Milner tucked his penalty down the middle after Mason Holgate had been adjudged to have hauled Lallana to ground inside the area.

It was a decision which left the young defender incensed, and he took his frustration out on Roberto Firmino five minutes later with a needless push which sent the Brazilian tumbling over an advertising hoarding and into the crowd.

Firmino reacted angrily and squared up to Holgate, but referee Bobby Madeley controversially kept his cards in his pocket following the skirmish despite a discussion with his fourth official.

There was no more goalmouth action to speak of before half time as Everton went into the break without having had a shot of note in the opening 45 minutes, although they would have been buoyed the news that Cenk Tosun completed his £27m move to Goodison Park during the interval.

It was Liverpool who looked most like adding to the scoreline in the opening stages of the second half, though, with Gomez squandering a golden chance from close range after being left unmarked at the back post from a corner.

The pressure continued to grow from Liverpool and Lallana came close a minute later when he latched on to Alex Oxlade-Chamberlain's through-ball, only to skew his effort wide of the target while off balance.

Jordan Pickford was then called into action to deny Andrew Robertson from a tight angle after the full-back had skipped past Yannick Bolasie and fired a powerful strike towards the roof of the net.

Debutant Van Dijk was the next to threaten when he rose highest to meet a free kick from Oxlade-Chamberlain shortly after the hour mark, but his header was straight at Pickford, who gathered at the second attempt.

Liverpool were made to pay for their missed opportunities shortly afterwards when the visitors levelled things up against the run of play, hitting the hosts with a quick counter-attack which was neatly finished off by Sigurdsson.

Bolasie and Ademola Lookman were both involved before Phil Jagielka found himself as the furthest man forward, and the Everton skipper kept his composure to cut inside and tee up Sigurdsson, who passed his finish into the bottom corner from the edge of the box.

Jagielka was proving to be an important figure at both ends of the field when he cleared a dangerous Firmino half-volley nine minutes from time, but there was nothing the veteran could do just three minutes later when Van Dijk ensured a dream start to his Liverpool career.

Pickford came to gather Oxlade-Chamberlain's corner into the box, but Van Dijk soared above the keeper and two other blue shirts before planting his header into the empty net in front of the Kop.

There was not enough time for Everton to muster a response in the closing stages, with a long-range Bolasie strike which flew over the crossbar the closest they came as their wait for victory over Liverpool extended to 16 matches - a new record for the Merseyside derby.

Everton are also still winless at Anfield in any competition this century and have now lost three games on the bounce, whereas Liverpool's unbeaten streak stretches to 17 games - their longest such run under Klopp.

LIVERPOOL (4-2-3-1): Karius; Gomez (Solanke, 77'), Matip, Van Dijk, Robertson; Can, Milner (Alexander-Arnold, 77'), Oxlade-Chamberlain, Lallana (Wijnaldum, 70'), Mane; Firmino

EVERTON (4-2-3-1): Pickford; Kenny, Holgate, Jagielka, Martina; McCarthy (Davies, 86'), Schneiderlin, Bolasie, Rooney (Lookman, 52'), Sigurdsson; Calvert-Lewin (Niasse, 82')