Marouane Fellaini went the closest to scoring as new Everton boss Roberto Martinez was kept waiting for his first win as they were held to a goalless draw by dogged West Brom.
The midfielder, subject of an approach from Manchester United this week, struck the post late on as Everton created the better chances of a tight Barclays Premier League contest at Goodison Park.
A Seamus Coleman cross-cum-shot also rattled the woodwork as the game entered injury time but the Baggies held on for a point in a lively finish.
Martinez will again feel frustrated after his side were held at Norwich in their season opener but there were plenty of encouraging signs for the Spaniard.
For West Brom, who also remain winless, there are more concerns with Steve Clarke's team looking toothless in the absence of striker Nicolas Anelka, who is on indefinite compassionate leave.
Everton always looked confident and only their lack of a killer touch in front of goal let them down.
The hosts pieced together their first promising move after six minutes when Steven Pienaar got a header on target from a Kevin Mirallas cross, although Ben Foster's save looked over-flamboyant.
Pienaar tested Foster again with a low shot from outside the area soon after and Ross Barkley, confidence no doubt swelled after last week's fine performance and goal against Norwich, drove across goal.
It took 25 minutes for West Brom to create a chance but Youssouf Mulumbu curled well wide from a Graham Dorrans cut-back.
Everton threatened again from a free-kick but Mirallas blasted into the wall and Barkley blazed well over.