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Brighton piled more pressure on Everton manager Rafael Benitez as the Seagulls climbed to eighth in the Premier League with victory in an entertaining match at Goodison Park.
Alexis Mac Allister scored a third-minute opener for the visitors following Neal Maupay's cushioned header and Dan Burn headed in when unmarked at the back post to double the lead after 28 minutes.
The returning Dominic Calvert-Lewin, absent through injury since August, had the chance to respond from the penalty spot after referee John Brooks reviewed Enock Mwepu's challenge on Anthony Gordon - but he fired his spot-kick over.
The half-time whistle was met by jeers from the home support but Gordon gave his side hope of a fightback with a deflected shot eight minutes after the restart.
Allan drew a superb save from Brighton goalkeeper Robert Sanchez as Everton chased an equaliser but Mac Allister's excellent second from the edge of the box put the away side back in control.
Gordon's second with 14 minutes remaining made for a tense finish, but he was unable to complete his hat-trick with a late chance and Everton remain 15th with defeat.
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Brighton's fine season continues as pressure increases on Benitez
It is one defeat in seven league games for Brighton, who followed up their draw against Chelsea at Stamford Bridge with this fine performance against Benitez's stuttering Everton.
The Seagulls had never previously won at Goodison but Mac Allister's early breakthrough, teed up by Maupay from Joel Veltman's cross, set the tone for a second victory in three games.
Slack defending allowed Burn to head in unchallenged after Mwepu flicked on Mac Allister's corner - and Calvert-Lewin's penalty miss ensured Potter's side had control at the break.
Everton's most recent game was 17 days ago, after matches against Newcastle and Burnley were postponed, and that 1-1 draw with Chelsea temporarily eased the pressure on Benitez.
The long-awaited return of Calvert-Lewin from injury is a huge boost for the Toffees boss, whose side managed only one win in 11 league matches in his absence.
Calvert-Lewin's previous appearance - and goal - came in a 2-0 win at Brighton on 28 August as the 24-year-old scored in three successive games at the start of the season, but he could not inspire his side on Sunday.
The England striker was unable to steer in Gordon's cross as he sought to make amends for his spot-kick miss, while Brighton threatened to wrap up the three points before the interval through Adam Lallana and Maupay.
Everton were much improved after half-time and gained a fortuitous lifeline when Gordon's strike deflected in off Lallana.
Gordon's performance offered a real positive on another disappointing afternoon for Everton and the lively 20-year-old responded again for his side from Jonjoe Kenny's cross after Mac Allister had scored a third for Brighton.
Despite their late resurgence, Benitez's side could not rescue a late point. Their 19 points from 18 games is the fewest they have had at this stage since the 2005-06 season.
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Everton
Squad number1Player namePickford
Squad number2Player nameKenny
Squad number22Player nameGodfrey
Squad number5Player nameKeane
Squad number4Player nameHolgate
Squad number23Player nameColeman
Squad number24Player nameGordon
Squad number16Player nameDoucouré
Squad number6Player nameAllan
Squad number11Player nameGray
Squad number9Player nameCalvert-Lewin
Squad number21Player nameAndré Gomes
Squad number33Player nameRondón
Brighton & Hove Albion
Squad number1Player nameSánchez
Squad number34Player nameVeltman
Squad number4Player nameWebster
Squad number33Player nameBurn
Squad number3Player nameCucurella
Squad number12Player nameMwepu
Squad number14Player nameLallana
Squad number8Player nameBissouma
Squad number10Player nameMac Allister
Squad number9Player nameMaupay
Squad number11Player nameTrossard
Squad number2Player nameLamptey
Squad number15Player nameModer
Squad number24Player nameDuffy
Line-ups
Everton
Formation 5-4-1
- 1Pickford
- 2KennyBooked at 9minsSubstituted forRondónat 76'minutes
- 22Godfrey
- 5Keane
- 4Holgate
- 23Coleman
- 24Gordon
- 16Doucouré
- 6AllanSubstituted forAndré Gomesat 87'minutes
- 11Gray
- 9Calvert-Lewin
Substitutes
- 12Digne
- 15Begovic
- 17Iwobi
- 21André Gomes
- 25Gbamin
- 33Rondón
- 50Simms
- 61Dobbin
- 62Onyango
Brighton
Formation 4-4-2
- 1Sánchez
- 34VeltmanSubstituted forDuffyat 61'minutes
- 4WebsterBooked at 63mins
- 33Burn
- 3Cucurella
- 12Mwepu
- 14LallanaSubstituted forLampteyat 61'minutes
- 8Bissouma
- 10Mac Allister
- 9Maupay
- 11TrossardSubstituted forModerat 81'minutes
Substitutes
- 2Lamptey
- 13Groß
- 15Moder
- 17Alzate
- 18Welbeck
- 20March
- 23Steele
- 24Duffy
- 30Richards
Live Text
Match ends, Everton 2, Brighton and Hove Albion 3.
Second Half ends, Everton 2, Brighton and Hove Albion 3.
Foul by Neal Maupay (Brighton and Hove Albion).
Michael Keane (Everton) wins a free kick in the defensive half.
Attempt blocked. Salomón Rondón (Everton) right footed shot from the left side of the box is blocked. Assisted by Ben Godfrey.
Attempt missed. Michael Keane (Everton) left footed shot from outside the box misses to the right.
Attempt missed. Salomón Rondón (Everton) left footed shot from the left side of the box is too high. Assisted by André Gomes with a cross following a corner.
Corner, Everton. Conceded by Shane Duffy.
Attempt missed. Anthony Gordon (Everton) right footed shot from the centre of the box is close, but misses to the left. Assisted by Salomón Rondón.
Substitution, Everton. André Gomes replaces Allan.
Foul by Enock Mwepu (Brighton and Hove Albion).
Mason Holgate (Everton) wins a free kick in the defensive half.
Substitution, Brighton and Hove Albion. Jakub Moder replaces Leandro Trossard.
Corner, Everton. Conceded by Shane Duffy.
Neal Maupay (Brighton and Hove Albion) wins a free kick in the defensive half.
Foul by Mason Holgate (Everton).
Foul by Alexis Mac Allister (Brighton and Hove Albion).
Ben Godfrey (Everton) wins a free kick on the right wing.
Corner, Everton. Conceded by Yves Bissouma.
Substitution, Everton. Salomón Rondón replaces Jonjoe Kenny.
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