Goals by Iliman Ndiaye and James Garner, both following passes by Jack Grealish, saw the Toffees to the sort of result their debut at Hill Dickinson Stadium demanded.
But Albion smoothed their path by wasting an open goal and a penalty - both by the returning Danny Welbeck - as well as various other promising opportunities, including a Matt O’Riley one-on-one.
They also hit the woodwork twice.
Welbeck started up front for Albion, which felt like a shrewd move.
But it was not all good news as Georginio Rutter missed out due to what was described as “a slight knock”.
The big news for Everton was a start for recent loan signing Grealish while striker Thierno Barry also came into the XI.
Everton’s fans and sound system were at impressive volume amid great scenes around the time of the players’ entry on to the pitch.
The hosts had early momentum and Ndiaye skipped past a weak tackle by O’Riley but his low shot was easy for Bart Verbruggen.
However, the noise soon abated, as the travelling supporters were happy to point out to their counterparts.
And Albion should have added to nerves among the locals by going ahead.
Firstly, through a piece of magic as Kaoru Mitoma scooped the ball over two defenders and, as it fell, thumped a volley over via the crossbar.
And then they were wasteful in the extreme on 19 minutes as Welbeck scooped a low cross from the troublesome Yankuba Minteh over an empty net.
The ball arrived slightly behind Welbeck but it was a chance he should have buried – and the sort of miss you always fear will come back to haunt you.
That it did just four minutes later as Grealish worked just enough room away from Wieffer to drive the ball across goal.
Max De Cuyper, an exciting attack-minded left-back who lost his man a couple of times in the draw with Fulham and got away with it, did so again and this time Ndiaye was unforgiving.
Still, though, there was unease in home ranks.
Jan Paul van Hecke hit the post via a deflection off Grealish from outside the box.
And, right on half-time, O’Riley was thwarted by Jordan Pickford when James Tarkowksi’s back pass sent him through on goal.
Albion ended the half with a barely credible deficit to make up and both their full-backs on yellow cards.
Their task became even tougher just seven minutes into the second half as Grealish played a simple pass and Garner arrowed a low shot into what looked an inviting gap between Bart Verbruggen and his near post.
Albion continued to prompt and probe in attack. Minteh was able at will to beat his man and then find an Everton boot with his delivery.
But it was the winger’s shot which offered a lifeline on 79 minutes.
Kiernan Dewsbury-Hall blocked with his arm but the spot kick by Welbeck was weak.
Pickford saved low to his left and what looked an ordinary Everton side could free-wheel to their fairy tale finale.
Everton: Pickford; O’Brien, Keane, Tarkowski, Garner; Iroegbunam (McNeil 71), Gueye; Ndiaye (Alcaraz 64), Barry (Beto 64), Dewsbury-Hall, Grealish (Armstrong 90+4); Barry. Subs: Travers, Chermiti, Aznou, Armstrong, Onyango.
Goals: Ndiaye 23, Garner 52.
Yellow card: Garner, Dewsbury-Hall, Beto,
Albion: Verbruggen; Wieffer (Gruda 85), van Hecke, Dunk, De Cuyper (Kadioglu 67); Baleba (Hinshelwood 46), Ayari (Gomez 67); Minteh, O’Riley, Mitoma; Welbeck. Subs: Steele, Veltman, Boscagli, Coppola, Milner.
Yellow card: De Cuyper, Wieffer, Kadioglu.
Referee: Stuart Attwell.