Salomon Rondon scored the decider in the 1-0 victory, nicking the goal off Jonas Olsson with the final touch right on the line, for his third in the last seven matches as the Baggies clinched three points with their only shot on target.
By contrast the home side threw everything at their opponents but there was far too little precision in what they did as of their incredible 34 attempts, only six were on target.
An indication of Everton's dominance - and at times desperation - was that over the course of 90 minutes every outfield Everton player, apart from their final, late substitute Leon Osman, attempted a shot.
The Baggies, lining up with four central defenders across the back, remained unmoved and unbowed, however, as they produced the sort of typically resolute performance for which their manager Tony Pulis is renowned.
On the back of three successive 3-0 victories (one in the FA Cup) Everton boss Roberto Martinez again named an unchanged side, meaning fit-again goalkeeper Tim Howard had to settle for a place on the bench.
However, for all the support for Joel Robles to be the new permanent first-choice, there were questions being asked of the Spaniard for West Brom's 15th-minute goal.
Stephane Sessegnon's inswinging corner was helped on by Olsson but Robles appeared to be convinced the looping ball was heading wide - it was not - and Rondon at the far post bundled it in on the goalline.
Everton have now conceded more goals at home (23) than any other side in Europe's top five leagues this season and is the main reason they have managed just four wins at Goodison this term.
What followed was effectively an advanced form of attack versus defence as Everton laid siege to the Gwladys Street goal with Ross Barkley and Tom Cleverley both hitting the woodwork within the space of three minutes.
The first half ended with the hosts having enjoyed 74 percent possession and had 17 shots, of which only two were on target, as Seamus Coleman (twice), Barkley, Romelu Lukaku, Ramiro Funes Mori and Aaron Lennon, among others, all had a go.
Lennon gave James Chester, out of his depth at left-back, a torrid time so it was no surprise the decision to replace him with Gerard Deulofeu 16 minutes from time was booed by a frustrated Goodison Park.
West Brom managed one attempt after the break when substitute Chris Brunt flashed a shot just wide of the far post but otherwise it was one-way traffic.
But with a lack of guile in the final third, coupled with West Brom's determination to throw everything in the way of every attack, Everton came up well short.