Cody Gakpo scored his first goal for Liverpool and Mohamed Salah was on target as Everton were consigned to a 2-0 Merseyside derby defeat at Anfield.
The Reds' hopes of securing a top-four finish had been rocked by a run of four top-flight games without a win, but they got their first Premier League victory of 2023 on Monday.
Salah rounded off a blistering counter-attack 13 seconds after James Tarkowski had struck the post with his 18th goal of the season, having failed to find the back of the net in his previous five matches, late in the first half.
Gakpo got off the mark early in the second half as Liverpool moved up a spot to ninth and kept their neighbours in the relegation zone by consigning boss Sean Dyche to a first defeat, with substitute appearances for the fit-again Diogo Jota and Roberto Firmino also providing boosts for Jurgen Klopp.
Netherlands forward Gakpo spurned a good chance to open the scoring when he headed wide from close range early in a frantic start.
Tarkowski came agonisingly close to scoring a second goal in as many games after his winner against Arsenal when he rose to meet Alex Iwobi's corner with a header that struck the post, and Liverpool were in front moments later.
Everton were badly caught out on the break as Darwin Nunez darted forward and picked out Salah, who tucked into an empty net with Jordan Pickford in no man's land nine minutes before the break.
The Toffees were once again caught on the break four minutes into the second half, when Conor Cody failed to deal with Trent Alexander-Arnold's dangerous cross and Gakpo opened his account with a tap-in.
Nunez was denied by Pickford from a tight angle and bent an effort just wide after being set up by Gakpo as Everton continued to be opened up all too easily.
Ellis Simms was withdrawn after being handed a start with Dominic Calvert-Lewin unfit, before Jota and Firmino were given great ovations when they came off the bench, and Tom Davies' glaring miss with a header from close range summed up Everton's day.