Albion had won their first away game for the past six seasons and only had themselves to blame for not making it seven on opening day at the Hill Dickinson Stadium.
They found themselves behind to a goal from Iliman Ndiaye at the break after hitting the woodwork twice and squandering two other chances in the first half.
Yankuba Minteh and Kaoru Mitoma were a constant menace down the flanks but no one could profit from their good work and when James Garner lashed in Evertonâs second in the 53rd minute Albion were up against it.
They were handed a lifeline with 13 minutes to go but Danny Welbeckâs poorly-struck penalty was saved by Jordan Pickford and Everton - not Albion - knew it was going to be their day.
A deafening noise greeted the teams just before kick-off as the home fans found their voices inside their new 53,000-seater home. But it wasnât long before the only sound was from 3,000 Albion fans in the north-east corner as their side took control.
Danny Welbeck's shot is blocked by Tim Iroegbunam. đˇ James Boardman.
They had two great chances to go ahead in the space of three minutes midway through the first half. Mitoma, so often a problem for Everton in the past, brilliantly brought Bart Verbruggenâs clearance under control before flicking the ball over James Tarkowski. His volley beat Pickford but came back off the bar.
Minteh then broke at pace into the box on the right but his low cross was fractionally behind Welbeck who scooped his shot from six yards over an empty net.
Kaoru Mitoma brings the ball under control before volleying against the bar seconds later. đˇ James Boardman.
Everton had offered little threat until the 25th minute when Jack Grealish got past Mats Wieffer and Ndiaye, arriving at the far post, was not marked as he side-footed the cross into the roof of the net.
Albion were denied by the woodwork again in the 39th minute. Jan Paul van Hecke shrugged off a challenge from Thiero Barry and curled his shot past Pickford, but against the far post. And in stoppage time came another great opportunity. Matt OâRiley dispossessed Tarkowski but Pickford saved at his feet as he tried to go round the Everton keeper.
Yasin Ayari runs at the Everton defence. đˇ James Boardman.
How Albion were made to regret those misses. Eight minutes into the second half Grealish laid the ball into Garnerâs path 20 yards out and his fiercely-struck low shot had too much power and pace for Verbruggen as it flew inside the left-hand post.
Albion frustrations were compounded in the 77th minute. Kieran Dewsbury Hall handled Mintehâs shot inside the box but Pickford guessed correctly, diving to his left to keep out Welbeckâs penalty. A decent performance by Albion except where it mattered most.
Jordan Pickford punches clear under pressure from Danny Welbeck. đˇ James Boardman.
The line-up
Verbruggen, Wieffer (Gruda 85), Dunk, van Hecke, De Cuyper (Kadioglu 67), Baleba (Hinshelwood 45), Ayari (Gomez 67), Mitoma, Minteh, OâRiley, Welbeck.
Subs not used: Steele, Milner, Boscagli, Veltman, Coppola.
Referee: Stuart Attwell
Attendance: 51,759