The high-stakes game at Anfield occasionally threatened to boil over
Struggling Everton made Liverpool fight for their win with a well-organised and disciplined performance that will give them hope they can still avoid the drop but was not enough to deny Jurgen Klopp's quadruple-chasing side.
Liverpool can find different ways to get the job done and after blowing Manchester United aside on Tuesday they demonstrated patience to finally break down Everton's stubborn resistance in what could prove to be a priceless win.
The deadlock was broken after 62 minutes when Andy Robertson arrived at the far post in front of the Kop to head home Mo Salah's cross.
Everton could not respond, although Demarai Gray powered a drive just wide, and Liverpool wrapped up the win when substitute Divock Origi tormented the neighbours once again when he headed in Luis Diaz's bicycle kick with six minutes left.
The loss means the Toffees end the day in the top-flight's bottom three for the first time since 6 December 2019. They were last in the relegation zone this far into the season in 1998-99.
This was not the flowing, blistering Liverpool that effectively won the FA Cup semi-final in the first 45 minutes against Manchester City at Wembley and who dismantled a hapless Manchester United at Anfield.
Everton's rearguard action and determination to disrupt Liverpool infuriated and frustrated Anfield in the first 45 minutes but eventually the dam broke and for all the visitors' courage they were not able to hold out.
Manager Klopp may well take as much pleasure from how Liverpool were forced to dig deep and overcome Everton's determined challenge as he would from the way they have swept so many teams aside this season.
Liverpool had to fight for control and, in truth, they were never in their normal level of command and still had to survive one or two anxious moments before the final whistle.
It puts the Reds right back in City's rearview mirror after Pep Guardiola's side thrashed Watford 5-1, and still on course for their ambition of adding the Premier League, Champions League and FA Cup to the EFL Cup.
Liverpool now face Villarreal in the Champions League semi-final first leg at Anfield on Wednesday as firm favourites and their challenge continues to gather huge momentum.
Everton find themselves in a very hazardous position in the relegation places after this loss at Liverpool followed Burnley's win against Wolverhampton Wanderers.
Frank Lampard's side are now in serious danger of relegation and their 17% possession displays how on the back foot they were, but there were other elements of this game that must encourage them as they try to avoid the drop into the Championship.
They will certainly believe they should have had a penalty when the outstanding Anthony Gordon tumbled under a challenge from Joel Matip with the score 0-0 after 53 minutes.
It looked like a clear penalty but was waved away by Stuart Attwell, who had earlier booked the talented youngster for diving.
Gordon also shot across the face of goal with a good chance before Liverpool took the lead and Gray fired a rising drive inches off target as his side looked to equalise.
Robertson also made a vital goalline clearance with Alex Iwobi poised to turn home Dele Alli's cross.
Everton were fiercely well-drilled until Liverpool went ahead. It was not pretty but Everton do not have to make excuses for that. They could hardly be expected to play an open, expansive game and make themselves lambs to the slaughter.
What is beyond doubt, though, is that Everton are now in serious peril, especially with Burnley seemingly completely reinvigorated after Sean Dyche was sacked and Mike Jackson appointed as caretaker.
Jackson has hauled in seven points in three games in charge and the momentum is now with Burnley. This is something Lampard and Everton must address, quickly.
-- Courtesy of BBC Sport