Richarlison gave the Blues the lead in the contest after three minutes as the hosts lost Jordan Henderson to injury during a frustrating first half.
Carlo Ancelotti's side then secured the three points when Gylfi Sigurdsson converted an 83rd-minute penalty.
Team news
Jürgen Klopp named an unchanged side to the one that beat RB Leipzig in the Champions League earlier on in the week.
Naby Keita started on the bench having been absent since December through injury.
First half
Everton came out of the blocks quickly and took a very early advantage in the encounter.
James Rodriguez played a through ball in behind the Liverpool backline, with Richarlison on the end of it to finish beyond Alisson Becker.
Liverpool responded well, though, and went close to equalising on three occasions within a two-minute period.
Roberto Firmino's side-footed finish inside the area was deflected by Michael Keane in the nick of time, before Henderson hit a first-time volley that was turned behind by Jordan Pickford.
Trent Alexander-Arnold's fizzing shot from distance was then tipped over by the Everton stopper as the Reds enjoyed a strong spell.
But they were dealt a blow when Henderson was forced off injured in the 29th minute – after initially trying to play on. Nathaniel Phillips was the captain's replacement.
Liverpool needed Alisson to push away a Seamus Coleman header from inside the box following Lucas Digne's ball in from the left.
Phillips then headed into the side-netting from an Andy Robertson out-swinging free-kick in the Reds' final opportunity of the first half.
Second half
Liverpool started the second period well, with Sadio Mane going close twice early on. The No.10 headed just fractionally over on the stretch before being denied by a last-ditch Keane challenge.
Klopp's side kept applying the pressure and Pickford was forced into another intervention.
After good link-up play with Firmino, substitute Xherdan Shaqiri slipped Mohamed Salah through on goal, with the Egyptian's close-range effort kept out by Pickford.
Having survived those chances, Everton then grabbed their second of the game from the penalty spot.
Following a counter-attack, Dominic Calvert-Lewin was thwarted by Alisson but then adjudged to have been brought down by Alexander-Arnold when following in on the rebound.
Sigurdsson stepped up and converted the spot-kick to give the visitors a two-goal lead and their first win at Anfield since 1999.