In Premier League news, an extraordinary first half saw the Everton go ahead after just 34 seconds, with Dominic Calvert-Lewin squaring for Abdoulaye Doucoure to tap in after Seagulls captain Lewis Dunk had been badly caught out.
Mali midfielder Doucoure added Everton's second just before the half-hour mark with a precise volley, into the right corner from Dwight McNeil's cross.
McNeil was then heavily involved as Sean Dyche's side extended their lead, driving towards the byeline and seeing his low cross deflect off the foot of Brighton goalkeeper Jason Steele into the back of the net.
A shambolic first period saw Brighton booed off at the interval, fortunate not to have suffered the embarrassment of being four goals down, with Dunk blocking James Garner's close-range effort on the stroke of half-time.
A mightily impressive performance from Everton, who repeatedly threatened on the break during the first half and defended stoutly in the second, saw them add their fourth when McNeil raced on to Alex Iwobi's incisive pass to round Steele and slot into an empty net.
Roberto de Zerbi's side, who were much improved after making four half-time changes, saw substitute Evan Ferguson (twice) and Alexis Mac Allister hit the woodwork, before they finally managed a scrappy consolation with Kaoru Mitoma's sliding effort rebounding off the post and bouncing in off Mac Allister.
A wonderful McNeil effort in added time, that flew into the top corner, ensured Everton had the final say as they moved out of the drop zone in style, to sit 16th in the table after leapfrogging Nottingham Forest, Leeds and Leicester City.
Many Everton fans travelled down to the south coast more in hope than expectation on the back of a seven-game run without a win.
In other Premier League news, they had also been hammered 4-1 by Brighton at Goodison in January when the Toffees' players completely fell apart. And prior to kick-off you would have had to trawl down to the eighth tier of English football to find a club with fewer league goals than Everton's 27 this season.
However, they found their clinical edge in sensational fashion as their supporters joyously celebrated in the stands.
Calvert-Lewin's turn away from Dunk set the tone in the build-up to the first goal, with the England forward delivering a fine display in only his fourth start in three months.
There were numerous others that excelled including Doucoure, who was out of favour before Dyche's arrival, and scored twice for the first time in the English top flight.
Aside from his two goals, the second of which involved wonderful composure, McNeil, who covered more ground than any other visiting player (12.76km) also made telling contributions when tracking back to help out his defence.
Nathan Patterson also largely nullified the threat of Kaoru Mitoma, while James Tarkowski and goalkeeper Jordan Pickford both made important interventions to keep the hosts at bay.
Pickford notably denied Republic of Ireland forward Ferguson on three occasions, brilliantly touching a header on to the post, athletically getting his fingertips to a rasping drive that clipped the crossbar on its way to safety and then saving by his left post late on.
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