Andy Robertson and Divock Origi headed Liverpool to within one point of leaders Manchester City as their second-half goals compounded Everton's relegation worries.
In a first half high on effort but low on scoring chances, the best opportunity fell to Everton's Abdoulaye Doucoure on 34 minutes, but he fired wide from Richarlison's pass.
Indeed, Liverpool had 82.7 per cent possession overall, but Everton's resolute defending frustrated the Reds as they failed to record a shot on target in the opening half at Anfield for only the second time this season.
Everton were close to breaking the deadlock after 56 minutes but Anthony Gordon dragged his shot wide of the far post.
Liverpool introduced Origi and Luis Diaz on 61 minutes and just 60 seconds later Origi's one-two with Mohamed Salah saw the Egyptian set up Robertson to head home. It was the defender's first Premier League goal at Anfield since September 2020.
Origi made the points safe with Liverpool's second goal five minutes from time, producing his sixth goal against Everton in all competitions with a header from Diaz's delivery.
Liverpool are second on 79 points, while Everton, without a single PL away point in 2022, lie third from bottom on 29.
The Toffees are two points behind 17th-placed Burnley with a match in hand over the Clarets.
Liverpool: 30 Apr v Newcastle (A)
Everton: 1 May v Chelsea (H)
Referee: Stuart Attwell. Assistants: Lee Betts, Timothy Wood. Fourth official: Jonathan Moss. VAR: Darren England. Assistant VAR: Dan Robathan.