Everton had actually kept Liverpool at arm's length before they were caught by a devastating counter-attack after 36 minutes, defender James Tarkowski's header hitting the post at one end before Darwin Nunez raced clear, squaring for Mohamed Salah to take advantage of goalkeeper Jordan Pickford's poor positioning with a simple finish.

Liverpool snuffed out any hopes of an Everton recovery four minutes after the break.

And it was a special moment for £45m January signing Cody Gakpo, as he tapped in Trent Alexander-Arnold's cross at the far post for his first goal since arriving at Anfield from PSV Eindhoven.

"The result is a massive relief," Klopp said to BBC's Match of the Day.

"The better you play the more likely you will score. The whole performance was extremely important for us because we needed to make a statement."

Liverpool show signs of life

It has been a miserable 2023 for Liverpool, encapsulated by heavy away defeats at Brentford, Brighton and Wolverhampton Wanderers, but the ease with which they were able to overcome Everton's admittedly flimsy challenge will lift the spirits.

This was a good night for Liverpool as manager Klopp was able to include Virgil van Dijk, Roberto Firmino and Diogo Jota on the bench after injury, the latter getting a rapturous reception when he was introduced as a second-half substitute for his first action since picking up the muscle problem that forced him to miss the World Cup in Qatar.

Firmino also got a late outing.

Another big plus was Gakpo's goal, a huge morale-booster for a player who has been quiet - although it would have been difficult for any new acquisition to come into such a struggling side who are ninth in the Premier League table, nine points adrift of the Champions League places.

Gakpo's confidence visibly lifted after his goal, showing assured touches and looking more at ease than at any time in his short Liverpool career.

"The dream is that you score in the first minute of the first game you play," said Gakpo. "In the end I have had to wait a little bit but I'm really delighted that I can score this one."

Another major plus for Klopp and Liverpool was the outstanding display of 18-year-old Stefan Bajcetic, who showed astonishing maturity in midfield with a comprehensive range of passing, a sure touch and a willingness to engage in the physical exchanges. It was a complete performance by the youngster.

Liverpool were nowhere near their best - they did not have to be - but the sight of Salah on the scoresheet, goals from trademark lightning counter-attacks and a clean sheet will change the mood around Anfield considerably before a crucial visit to Newcastle United on Saturday night.

Liverpool's victory was their first in the Premier League since 30 December

Everton revert to type

Everton started the Dyche era with a performance full of passion and competitive edge to stun Arsenal, but this was a reminder of why they face a serious relegation battle, and evidence that the display against the Gunners was very much the exception in this season of struggle.

None of the things Dyche expects from his teams was on show here. They were passive, slow and careless in possession - their major flaw once again exposed as they barely mounted a threat apart from Tarkowski's header that actually resulted in Liverpool breaking the deadlock.

The absence of Dominic Calvert-Lewin, injured once more after figuring for an hour against Arsenal, underscores the folly in not signing a replacement for Richarlison, an error in the summer compounded by the failure to correct it in January.

Much rests on the unreliable fitness of Calvert-Lewin, with Dyche having to rely on 22-year-old Ellis Simms, making only his second Premier League start. The youngster, understandably, struggled on his own up front with no service.

Everton now have a vital home game against managerless fellow strugglers Leeds United on Saturday and Dyche will want more of what they showed against Arsenal and less of what they produced at Anfield.