Alexis Mac Allister scored twice as Albion made a brilliant start to the year with their first ever win at Goodison Park.
The midfielder scored Albion’s quickest-ever Premier League away goal inside three minutes and Dan Burn got his first of the season with a close-range header midway through the first half.
After Dominic Calvert-Lewin missed a penalty, Anthony Gordon pulled one back early in the second half but Mac Allister made it 3-1 with a thunderous finish into the top corner from outside the box.
Gordon scored his second with 14 minutes to go but Albion dug in to claim a thoroughly deserved first win at Everton at the tenth attempt.
Graham Potter made three changes to the team that drew at Chelsea. Adam Webster, Leandro Trossard and Enock Mwepu came in with Jakub Moder, Tariq Lamptey and Solly March dropping to the bench.
It took Albion just three minutes to stun Goodison Park with a superbly worked goal. Trossard’s cross-field pass found Veltman, who was making his 50th appearance for the club, in space on the right. His measured centre was cushioned into the path of Mac Allister who used his right instep to volley past Jordan Pickford. At 2 mins 43 secs it was Albion’s third fastest Premier League goal.
Jonjoe Kenny picked up a yellow card for a foul on the goal scorer and there was nervousness among Everton fans every time Albion attacked. Their mood got worse in the 21st minute when Burn made it 2-0 with Albion’s fourth set-piece goal of the season.
Mac Allister’s corner was flicked on by Mwepu and Burn, who had peeled into space at the back post, headed home from three yards. Bissouma, who became the seventh Albion player to make 100 Premier League starts, deserved credit for winning the corner after a brilliant 30-yard run and shot, which was deflected behind.
Everton had a fortunate way back into the contest three minutes later. Mwepu made contact with Gordon just inside the box and after initially waving Everton appeals away referee John Brooks gave a penalty after a VAR review, but Calvert-Lewin’s spot-kick brushed the top of the bar and into the stand.
Pickford had to scramble to his right to push out Adam Lallana’s shot from 20 yards before holding Burn’s header from the corner; There were a couple of scares in the Albion box but they had another clear sight of Pickford’s goal before the break. Lallana and Trossard cleverly created space for Maupay whose effort from just outside the box curled a yard over the target.
So far so good, but Everton were back in it six minutes after the break when Gordon cut in from the right and his left-foot shot took a hefty deflection off Lallana, completely wrong-footing Sanchez who could only watch as the ball went into the bottom right-hand corner as he dived the other way.
The mood around Goodison was transformed. When the hosts won a corner they worked the ball to Demarai Gray whose shot swerved in the air forcing Sanchez to punch it away.
Albion brought on Tariq Lamptey and Shane Duffy after an hour with Veltman and Lallana making way, Lamptey to counter Gray’s increasing threat down the left in a 3-5-2.
Sanchez extended his frame to flick Allan’s 25-yarder over the bar as Everton pushed but they were stunned on 70 minutes by a goal of the season contender from Mac Allister. Maupay’s ball from the left was laid back by Mwepu and the Argentinian struck it first time from just outside the D into the top right-hand corner, his fourth goal of the season. It also marked the first time Albion have scored three or more goals in a Premier League match this campaign.
But if Albion thought they had the game won they were wrong. Six minutes later the unmarked Gordon steered Kenny’s low cross from the right into the bottom corner having started the move himself. Everton were back in it.
Gordon could have equalised with two minutes left but stabbed the ball wide from close range. Substitute Salamon Rondon blasted the ball over By then both sides were out on their feet but Albion dug in and are up to eighth in the table.
Albion: Sanchez; Veltman (Duffy 61), Webster, Burn, Cucurella; Mwepu, Bissouma; Lallana (Lamptey 61), Mac Allister, Trossard (Moder 81); Maupay.
Subs: Steele, Alzate, Moder, Gross, Welbeck, March, Richards.
Referee: John Brooks