Twice Everton came close to reeling in Brighton & Hove Albion but twice they were unable to gain parity.
Anthony Gordon, Everton’s best player, scored his first goal for the Club after 53 minutes to cut a deficit that stood at two.
By the time Gordon popped up with his second, crowning a move that was all about the forward’s desire and mobility, Everton had conceded what would ultimately be a decisive third goal.
Alexis Mac Allister crashed in that ‘winner’ – on 71 minutes – the Argentine adding to his tidy third-minute finish.
Dan Burn headed Brighton’s second before Dominic Calvert-Lewin, playing for the first time in more than 18 weeks, lifted a penalty over the bar after Gordon was fouled by Enock Mwepu.
Gordon sped infield to thump a low effort that took a heavy deflection off Adam Lallana for 2-1 eight minutes after the restart. And the player from Everton’s Academy surged onto Jonjoe Kenny’s delivery to turn in his second after 76 minutes.
Everton had a handful of half-chances to equalise but were denied by a deadly combination of good goalkeeping, stoic defending and wayward finishing.
They stay 15th, six points behind 10th placed Leicester City, who visit Goodison Park in nine days time.
Everton were up against it from very early on and two goals behind before the mid-point of the opening half.
There was never a sense, however, of Brighton being entirely on top of their task of resisting some form of Everton comeback. Certainly, the Evertonians, stayed right with their team, offering a customary, stirring Goodison mix of raw encouragement and howls of indignation every time referee John Brooks whistled the ‘wrong’ way.
Not many appeals for a penalty when Gordon was buffeted by Mwepu after 22 minutes, mind.
Not that the final decision was incorrect, far from it. It was just that Everton’s attempt for a quick response to falling two goals behind appeared dead when Joel Veltman halted Demarai Gray’s progress close to goal.
As Gordon tried to retrieve the loose ball he was knocked off balance by Mwepu, an infringement detected by VAR Stuart Attwell.
Calvert-Lewin took the penalty, three minutes after the initial foul, but lifted his effort skywards.
The Everton striker, treated to a hero’s return from all four sides of Goodison – the cheers when his inclusion was confirmed were surpassed only by the pre-game ovation for new signing Vitaliy Mykolenko – thought he was set for an opportunity in the opening minutes. That was until Burn timed his slide to intercept Gray’s low pass, aimed at Calvert-Lewin in the penalty area.
Towering centre-half Burn won a wrestle with Calvert-Lewin when Seamus Coleman, deployed as a wing-back on the left, swung in a cross.
Indeed, Calvert-Lewin had a shadow in the shape of the angular Burn pretty much from start to finish.
Burn, with his height and physicality, is a nuisance for opponents in both penalty areas.
And on 21 minutes, he escaped at the back post to meet Mwepu’s flick on a left-wing Mac Allister corner and bullet a header into the net.
Burn’s set-piece goal provided a cushion for the visitors, who hit the front with a neatly-worked third-minute strike.
Veltman arrowed in the cross from the right, Neal Maupay arching his back to take the sting off his header and redirect the ball into the path of the fast-advancing Mac Allister.
The Argentine arrived in a blur but had the sureness of mind to take stock of his situation and place a volley beyond Jordan Pickford.
Mason Holgate blocked a low drive from Leandro Trossard as Brighton tried to capitalise on their bright opening.
Everton for their part made a couple of forceful challenges in a bid to alter the mood. Kenny went into the book for a tackle on Mac Allister but the sight of the Everton player aggressively pursuing possession had the desired effect of whipping up teammates and supporters alike.
Steadily, Gray and Gordon began imposing themselves on the flanks, the interchanging Everton forwards plainly having the beating of their full-backs.
Yves Bissouma resorted to foul means to stop one Gray dash across field but the Everton player found his range with a number of low deliveries.
Burn thrashed one Gray cross clear. Gordon couldn’t control another, steering wide of the near post after darting inside from his post on the left.
Bissouma was responsible for winning the corner from which Burn converted – the Malian gliding forwards with the ball stuck to the end of his toes, before a shot that went behind off Allan.
Everton prodded and jabbed without landing a meaningful blow for the remainder of the opening 45 minutes.
Robert Sanchez took off to punch clear of the looming Gordon after Coleman forced a ball forward from about 18 yards.
And when Gordon shrugged off Mwepu to hang up a deep cross from the left, Calvert-Lewin stretched to keep play alive, only to look askance when the dropping ball fell for Marc Cucurella to clear.
Brighton, with their fast and clever passing and mobile attackers, retained a sizeable threat whenever breaking out of their defensive shape.
Pickford displayed quick feet to get across his goal and turn around a bouncing Adam Lallana strike and the elusive Maupay wriggled free for a 20-yard shot that whizzed over the top.
Burn began the second half much like he started the first, intervening decisively when Calvert-Lewin turned a Gray delivery goalwards.
But Everton drew within sight of the away side when Gordon got maximum reward for his bravery, the patently crestfallen Lallana deflecting the 20-year-old’s seemingly optimistic attempt past a visibly furious Sanchez.
Calvert-Lewin couldn’t make clean enough contact to steer Kenny’s cross on target soon after and Sanchez needed to keep his eye on the ball when Gray sent a fizzing drive swirling towards the Brighton keeper.
Allan, who supplied the looping pass for Gordon to go on and score, was the width of Sanchez’s fingertips from scoring his first Everton goal on 68 minutes.
The Brazilian, on his 40th Premier League appearance, took aim from the thick end of 30 yards.
Sanchez, though, leapt from right to left to make the faintest of contact with his right hand and concede a corner.
The direction of travel appeared firmly in Everton’s favour until Mac Allister inflicted a hammer blow.
Holgate executed a last-man tackle of Maupay, seemingly stemming an increasingly rare Brighton attack.
Burn was first to Michael Keane’s subsequent clearance to feed Trossard on the left.
Trossard steered play back to the 18-yard line, where Mwepu applied a gossamer back heel.
The ball sat up for Mac Allister on the edge of the ‘D’ and he cracked it into Pickford’s top-left corner.
Everton fell flat for a couple of minutes, visibly rocked by the setback.
Gordon, though, was a man on a mission and he started and finished a slick move for his second goal.
The Englishman retreated to find Calvert-Lewin, who ushered in Kenny down the right. Gordon continued his run and was on the spot to rifle in at the front post from Kenny’s delivery along the floor.
Kenny immediately made way for Salomon Rondon as Rafa Benitez switched to a back four, with the substitute joining Calvert-Lewin in a two-man attack.
Rondon was the provider, however, when Gordon toed a hat-trick chance past the upright with two minutes remaining.
Sixty seconds later and Rondon flashed over with an effort on the full following a cross from fellow substitute Andre Gomes.
There was time for Keane to have a go from distance, too, but his radar was off and Brighton held on.