Marouane Fellaini and Leighton Baines both played from the start for the Toffees despite Manchester United's public interest but neither side could provide a spark
Both sides had chances of note in a game largely defined by an entertaining tustle between the teams' respective midfields but Marouane Fellaini could have given his side the points late on at Goodison Park only for his close-range effort to bounce back off the post. A Seamus Coleman cross-cum-shot also hit the bar in injury-time as Everton were left frustrated.
The draw represented the second such result for the hosts in as many games in the new season, while West Brom got off the mark following their disappointing last-minute loss to Southampton in their first game.
Having been pleased with his side's efforts in an opening-day draw with Norwich, Roberto Martinez named an unchanged team. Steve Clarke, meanwhile, made just one alteration despite his side's disappointing 1-0 defeat to Southampton, as Nicolas Anelka - who has been allowed leave on compassionate grounds - was replaced by Markus Rosenberg.
Fans were treated to an entertaining start at Goodison Park, as both teams exchanged counterattacks in a frenetic opening five minutes. Everton had the best of these chances, with Steven Pienaar's goalbound stooping header from a Ross Barkley cross bringing a fine save from Ben Foster, and the South African forced the 30-year-old into action again soon after with a low left-footed drive.
Those two efforts preceded a spell of domination for the hosts, as they displayed the expansive style that new boss Martinez is attempting to instil during a period of possession. West Brom arguably created the best two chances of that stint, however, as Youssouf Mulumbu curled the ball narrowly wide before James Morrison's half-volley dipped just over the crossbar.
Everton's frustrations over their inability to grab a breakthrough boiled over just before half-time, though, when Leon Osman jumped into a needless challenge with Rosenberg for which he was fortunate to avoid a booking. But that moment of controversy did not push either side on to greater heights, as the half petered out without either finding the opening goal.
The battle of wills in midfield continued into the early stages of second half, though West Brom came close to finally breaking the deadlock when Graham Dorrans' inswinging free kick was hacked away at the back post by Nikica Jelavic. Everton then took their turn to go close as Foster dived low to his right to keep out Barkley's long-distance effort before getting up well to stop Fellaini smashing the rebound home.
The topsy-turvy nature of the encounter did not let up as time progressed and West Brom again went close just past the hour mark when Tim Howard did well to keep out a driven effort from Morrison. But the introduction of Matej Vydra and Arouna Kone around the 70-minute mark came as no surprise, with goals proving elusive for both teams.
It was a striker who had started the match who went closest next, however, with Shane Long's low left-footed effort being well kept out by a diving Howard in the Everton goal. The hosts were offered some hope in the 78th minute, when the seemingly unbeatable Foster made way for substitute shot-stopper Luke Daniels after picking up an injury.
And the 25-year-old was in fact lucky not to concede in the immediate aftermath of his arrival. Fellaini did well to control a Phil Jagielka diagonal pass into the box on his chest but his poked finish came back off the inside of the far post and was desperately cleared by a sliding Jonas Olsson.
Daniels was called on again with just three minutes of normal time remaining as he got down to stop a deflected Seamus Coleman strike and had to watch the Irishman's cross onto the bar but both teams were forced to settle for a point after failing to grab a crucial goal in the final moments.
More to follow...