The Seagulls went into the match in better form than their hosts and sitting above them in the standings on goal difference, but they were second best at Goodison Park as an own goal from Gaetan Bong and Cenk Tosun's first home goal for Everton saw the Toffees ease the pressure on manager Sam Allardyce.
Things only got worse for the visitors when Anthony Knockaert was shown a straight red card for a rash challenge on Leighton Baines, although they avoided an even heavier defeat courtesy of Mathew Ryan's late penalty save to deny Wayne Rooney.
The result lifts Everton up to ninth in the table and now within one win of the 40-point mark, whereas Brighton drop down to 11th place but remain seven points clear of the relegation zone.
Allardyce welcomed Rooney and long-term injury absentee Baines back into his starting lineup, and the hosts got off to the brighter start with a chance after only eight minutes when Phil Jagielka rose highest from a corner, only to nod his effort over the crossbar.
The Toffees continued to see the lion's share of possession throughout the first half, but clear chances were difficult to come by and neither side came close to finding the breakthrough until midway through the opening 45 minutes when Ryan missed a corner, allowing Yannick Bolasie to stick out a leg and divert the ball narrowly wide of the target.
Brighton's first sight of goal did not arrive until the 25th minute, although it brought about the only shot on target of the first half as the in-form Glenn Murray stung the palms of Jordan Pickford with a rasping drive from distance.
Ryan was called into action to turn Gylfi Sigurdsson's wicked cross away from goal, although his parry only took the ball as far as Bolasie, who sliced his follow-up volley over the crossbar.
Bolasie continued to look the most likely for the home side and came close again 10 minutes before the interval when he anticipated a flick-on but steered his header wide of the target from a difficult angle.
For all of their possession, Everton failed to truly test Ryan in the first half, and the hosts made a sloppy start to the second half too as Brighton saw the majority of the ball in the opening exchanges after the interval.
However, just as it looked as though control of the game may be slipping away from them, Everton broke the deadlock on the hour mark when Bolasie was allowed to cut inside and deliver a cross from the left which Bong steered into his own net while under pressure from Theo Walcott.
A second almost arrived in similar fashion moments later when again Bolasie's cross caught Bong out, but this time Walcott's touch back into the middle was pushed away by Ryan.
Chris Hughton turned to record signing Jurgen Locadia off the bench in an attempt to turn things around, and the sub almost had an immediate impact when the ball broke to him in the box, only for Seamus Coleman to react quickly and make a crucial block.
Everton were soon able to effectively take the game away from their visitors, though, when Baines played a one-two with Dominic Calvert-Lewin before sending a pass inside for Tosun, who slammed a powerful effort in via the underside of the crossbar for his first goal at Goodison Park.
Brighton did almost haul themselves back into the contest in the most spectacular fashion when Pascal Gross spotted Pickford off his line and tried his luck straight from kickoff, with his halfway-line effort floating just past the post.
Any lingering hope was well and truly extinguished just a minute later, though, as Knockaert flew into a dangerous two-footed challenge on Baines, who was fortunately able to dodge any major impact from the reckless Brighton midfielder.
The challenge was still bad enough for referee Roger East to produce a straight red card, though, ruling Knockaert out for the next three matches when Brighton will be bidding to keep themselves clear of the relegation battle below them.
It could have been even worse for the Seagulls when Calvert-Lewin was sent tumbling in the area late on, but Ryan kept the deficit at two with a stunning penalty save to deny Rooney.
Indeed, that almost allowed Brighton to force a nervy ending to the match when a loose Mason Holgate pass released Leonardo Ulloa in the first of four minutes stoppage time, but Gross's eventual strike was parried away by Pickford as Everton avoided any late scares.
The Toffees now sit 10 points clear of the relegation zone courtesy of a third consecutive home win, whereas Brighton's seven-match unbeaten streak across all competitions - which included three consecutive wins prior to today's match - comes to an end.
EVERTON (4-3-3): Pickford; Coleman, Jagielka, Keane, Baines; Sigurdsson, Davies (Klaassen, 84'), Rooney; Walcott (Calvert-Lewin, 73'), Bolasie (Holgate, 78'), Tosun
BRIGHTON (4-4-1-1): Ryan; Schelotto, Duffy, Dunk, Bong (Suttner, 82'); Knockaert, Kayal (Locadia, 69'), Propper, Izquierdo; Gross; Murray (Ulloa, 77')