Albion went into the match looking to record a fourth successive top-flight win for the first time since 1981 and for just the second time in their history.
Despite Lewis Dunk equalising in the first half, two goals from Richarlison and Seamus Coleman's first goal for two years gave the Toffees all three points.
Chris Hughton named an unchanged starting line-up for the third successive match, while Everton boss Marco Silva also named an unchanged line-up from their 2-1 defeat at Manchester United last week. That loss ended the Toffees' run of three consecutive league victories.
The hosts started brightly and Gylfi Sigurdsson wasted a glorious chance to give them an early lead when he was left unmarked in the penalty area 12 yards out but shot Seamus Coleman's cut-back across goal on four minutes.
After Dunk was booked for dragging back Richarlison, Sigurdsson then missed another decent opportunity when he side-footed into the side netting from ten yards, with half of Goodison Park thinking the ball had gone in.
Chances continued to come the home side's way and Kurt Zouma headed a Lucas Digne corner wide on 18 minutes, before Glenn Murray bundled the ball wide at the other end after Dunk headed a free back into the danger area.
Everton went ahead on 26 minutes when they broke quickly from a Brighton corner and after Sigurdsson played a one-two with Bernard, the Iceland midfielder slipped the ball through for Richarlison to rifle home a lovely team goal from eight yards.
Against the run of play, Albion levelled seven minutes later. Alireza Jahanbakhsh played a corner short to Solly March and his cross was powerfully headed home by Dunk from close range for his first goal of the season.
The Seagulls then got a foothold for the remainder of the half as the sides went in level at the break.
Like the first half, Everton made a quick start after the break.
Idrissa Gana Gueye was unlucky to see a shot from the edge of the area come back off a post on 48 minutes and Theo Walcott then volleyed over from a couple of yards when Bernard put the ball back across goal.
The Toffees did go back ahead just two minutes later when Coleman drilled home a low 20-yard strike from the edge of the penalty area.
Michael Keane headed over a Sigurdsson free kick just before the hour and the hosts continued to look the more threatening but few chances came at either end, despite Hughton sending on Anthony Knockaert and Yves Bissouma.
Everton moved 3-1 ahead 13 minutes from time when Richarlison cut out Dunk's pass, burst forward, rounded Ryan and tapped home into the empty net for his second goal of the match.
Summer signing Florin Andone was immediately introduced to make his Brighton debut and Jahanbakhsh saw a 20-yarder pushed over by Jordan Pickford but Everton saw out the game comfortably to return to winning ways.
EVERTON: Pickford; Coleman, Keane, Zouma, Digne; Gomes, Gueye; Walcott (Calvert-Lewin 85), Sigurdsson (Mina 89), Bernard (Lookman 68); Richarlison. Subs: Steklenburg, Baines, Tosun, Davies.
BRIGHTON: Ryan; Bruno, Duffy, Dunk, Bong; Jahanbakhsh, Kayal (Andone 77), Stephens, Izquierdo (Knockaert 63); March (Bissouma 70); Murray. Subs: Steele, Locadia, Balogun, Bernardo.
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