Andy Robertson's second goal of the season produced the vital breakthrough after 62 minutes of dogged resistance from the visitors and Divock Origi enhanced his derby legacy with his sixth goal in nine cross-city matches.
This was far from the humiliation many predicted – and Toffees fans feared – as Everton showed they can be organized and put up a fight but the one thing manager Frank Lampard could not bridge was the gulf in class.
Victory reduced Liverpool's gap to Premier leaders Manchester City back to one point, while leaving their near neighbours now two points from safety in the bottom three.
Everton's tactics were understandably evident from the off: get behind the ball in a 4-5-1, eat up as much time as possible at restarts and frustrate the life out of their opponents.
It could have been an approach straight out of Lampard's former mentor Jose Mourinho's playbook, who famously came here in 2014 with Chelsea to derail a Reds title bid.
"We can't fault our efforts. It is a tough place to come," Everton midfielder told Sky Sports.
"It (the game plan) was just to make it difficult for them, to throw them off a bit and hopefully take our chances, but unfortunately they scored two goals."
Richarlison, who was frequently on the floor and ate up a lot of time for treatment, and goalkeeper Jordan Pickford, who collapsed to the floor like a weary marathon runner every time he collected the ball in two hands, drew most of the ire from Anfield.
"In the first half, we were not at our best and started playing their game instead of concentrating on our own," Robertson said.
"We had to focus on our own game, playing football, being patient ... we created chances and managed the second half better. Patience was key."
Incredibly, Everton had the better chances early in the second half as Gordon and Alex Iwobi both directed attempts wide.
Liverpool sent on Luis Diaz and Origi just before the hour and the latter, who has made a habit of making his own personal derby history, combined with Mohamed Salah for Robertson to head home unmarked at the far post.
The deluge looked like it would start with Diaz and Salah, who volleyed over, having chances before Demarai Gray whistled a shot past Alisson Becker's right-hand post to show the visitors were not yet done.
Origi, who has barely played this season, finished things off with a close-range header five minutes from time.
The party rolls on at Anfield, with a Champions League semi-final first leg at home to Villarreal on Wednesday, but things do not get any easier for Everton with Chelsea up next at Goodison at the weekend.
