An own goal from Gaetan Bong on the hour mark broke the deadlock in what had been a tight affair up to that stage, but from then on Everton were on top and Cenk Tosun added a second with just under 15 minutes remaining.
Brighton were then reduced to 10 men when Anthony Knockaert's reckless challenge was punished by a straight red card, but Mathew Ryan prevented things getting even worse for the visitors with a late penalty save to deny Wayne Rooney.
Find out how all of the action unfolded courtesy of our minute-by-minute coverage below.
These two sides are only separated by goal difference in the table as things stand, but neither are yet safe from relegation and victory today would take them one step closer to that.
There is plenty riding on this one, then, so let's waste no time in getting started with a look at the team news...
Well, we'll start with the hosts, who notably welcome Wayne Rooney back into their starting lineup this afternoon after the former Manchester United man was dropped against Burnley last weekend.
Rooney looks like he will play in a deeper-lying midfield role, with the trio of Walcott, Sigurdsson and Bolasie in front of him, although Sam Allardyce could also deploy a 4-3-3 system with the players he has chosen today.
Walcott made a bright start to life at Goodison Park but has gone off the boil a little in the last few weeks, whereas on the other side Bolasie is still perhaps working his way back to full fitness after such a long time out. There is plenty of pace on those wings for Everton today, though.
Calvert-Lewin drops to the bench as a result, being replaced by Bolasie, while the Toffees will also be without the influential Idrissa Gueye in midfield as he is replaced by the returning Rooney.
Ashley Williams will serve the first of his three-game ban for his senseless red card against Burnley last weekend, but captain Phil Jagielka makes a timely return to fitness to fill in for him.
Indeed, it is an unchanged back five from that famous win over the Gunners as Duffy, Dunk and Bong also continue in front of Mat Ryan. Dunk, of course, scored against Arsenal and will be hoping to threaten at the sharp end of the field again today.
Murray already has seven Premier League goals this calendar year - only Mohamed Salah and Sergio Aguero have managed more in the entire league - and in all competitions his recent record stands at eight goals in his last 10 outings.
Gross has registered eight Premier League assists already this season, and only Manchester United's Paul Pogba and Man City trio Kevin De Bruyne, Leroy Sane and David Silva have managed more.
That change comes in midfield as Stephens drops out of the starting lineup, with Kayal replacing him in the middle of the park. Knockaert, Propper and Izquierdo do all retain their places in midfield, though, as Brighton stick with their 4-4-1-1 formation.
The Toffees expected to be pushing for the top six this season considering their summer investment, but with nine games of the season remaining they are in the bottom half of the table and 11 points of sixth-placed Arsenal - not to mention 24 adrift of the top four.
Everton should have done enough to avoid relegation this season, but they do still have a bit of work to do in these closing months with two wins still needed to reach the magic 40-point mark.
In many ways Allardyce was fighting from the off as many fans did not want him in the first place due to his style of football, and now that they have suffered a prolonged dip in form there have been serious questions raised over his long-term future.
Across all competitions the Toffees have now lost three of their last four and only won two of their last 12, losing seven of those since December 18 - including each of their last two.
It will get a lot tougher for the Toffees after that as they then face Manchester City and Liverpool, although both of those games do come at home, where they have been much more impressive this season.
If only home form counted Everton would be sitting pretty in seventh places having amassed 26 of their 34 points at Goodison, with only the top six having picked up more home points this season.
At home Everton have scored more than twice as many goals as they have away, in addition to conceding significantly fewer too. A good omen for this match is also the fact that they have won their last six league home games against newly-promoted teams, and another victory today would give them a run of seven for the first time ever.
Like Everton, there is still work to do for the Seagulls, but if you had offered them ahead of their first season in the Premier League the chance to be sitting in the top half of the table with nine games of the season remaining, they would have snapped your hand off.
Brighton still need two wins to reach the coveted 40-point mark, but with seven points separating them from the bottom three and only nine games remaining, they should be relatively safe barring a big collapse.
That said, Brighton are coming into this match off the back of a big win over Arsenal - their first ever Premier League triumph over one of the 'big six'. It was a deserved win too, with Dunk and Murray helping to pile more misery on the Gunners.
The Seagulls' unbeaten run now stretches back seven games, with five wins in that time stretching back to January 20. It is a brilliant run of form for a team who expected to be fighting for their lives until the very end, and another win here could just be enough to see them safe.
You have to go back to May 1981 for the last time Brighton won three top-flight games on the bounce, while even a draw today would see them go six top-flight games unbeaten for the first time since November 1981.
Chief among those is defensively - an area where Brighton have been good for most of the season but could still do better. Only Burnley have conceded fewer goals outside the top five this season, but Brighton are also still without a clean sheet in the Premier League this calendar year, failing to keep one in any of the eight games since their goalless draw with Newcastle on December 30.
In the Premier League Brighton have gone eight away games without a win, picking up only three points from the 24 on offer, and stretching back to their last spell in the top flight they have won just two of their last 21 away outings at this level.
Indeed, only Everton, Stoke and West Brom have picked up fewer points on their travels this season, while no team has managed fewer goals than Brighton's seven.
The formbook points to a Brighton win today, and they will certainly be full of confidence heading into this match, but it is hard to ignore their dreadful away record, whereas Everton have been much better at home. I'm going for a home win, despite Everton's recent troubles.
SPORTS MOLE SAYS: Everton 2-0 Brighton
In the league there have only been nine meetings in the past, with Everton winning four of those, with four draws and just one defeat in that time - the solitary loss coming in February 1982 when Brighton ran out 3-1 winners at home.
Most recent could be fairly misleading there, though, as the last of those came in October 1982, with Everton were held to a 2-2 draw by Brighton here at Goodison.
Anthony Knockaert thought he had done enough to give Brighton all three points with his 82nd-minute opener, but Wayne Rooney rescued a point for the Toffees with a stoppage-time penalty.
EVERTON STARTING XI: Pickford; Coleman, Keane, Jagielka, Baines; Rooney, Davies, Walcott, Sigurdsson, Bolasie; Tosun
BRIGHTON STARTING XI: Ryan; Schelotto, Duffy, Dunk, Bong; Knockaert, Kayal, Propper, Izquierdo; Gross; Murray
Everton have dominated the possession with Brighton content to sit back, but clear-cut chances have been very few and far between at both ends of the field. Everton have looked the most likely to break the deadlock, but it is Brighton who registered the only shot on target of that first half.
The ball fell to the striker around 25 yards from goal and he only had one thought in his mind as he released a powerful effort which stung the palms of Pickford. It was a fairly routine save for Pickford, and one he would expect to make, but it was still a decent hit from Murray.
Bolasie has looked the most likely to break the deadlock having threatened on three separate occasions so far today, while Sigurdsson drew what is the closest we have seen to a Ryan save so far with a wicked cross that needed to be palmed away.
It is really poor defending from Brighton as they allow Bolasie to cut inside from the left flank completely unopposed. The winger has the time and space to pick out a cross to the back post, where Walcott has peeled off Bong. Bong is too relaxed when looking to deal with the danger, and the final touch may well come off the defender last as Walcott looks to volley it home.
It is a first Goodison Park goal for Tosun here as Baines and Calvert-Lewin play a one-two before the former picks out Tosun in space just inside the box. There is an element of fortune about his finish as it takes a deflection before crashing in off the underside of the crossbar.
The Brighton man was caught with a ball to the head moments prior, and then he dives in two footed, shin high on Baines. There isn't much contact, but the red card is the right decision.
It could have gone either way until a Gaetan Bong own goal broke the deadlock on the hour mark, but from then on Everton deserved their win, with Tosun adding a second before Knockaert was shown a straight red card for the visitors.
It could have been even worse for Brighton too as Everton were awarded a late penalty, but Rooney saw his spot kick very well saved.
Thank you very much for joining Sports Mole for this match as Everton leapfrog Brighton in the table courtesy of a 2-0 win at Goodison. I will leave you with our match report, and be sure to stick around for reaction too.
From me, though, it is goodbye for now!