Marouane Fellaini came closest to breaking the deadlock with a shot against the post, while Seamus Coleman also hit the crossbar late on.
West Brom goalkeeper Ben Foster pulled off a number of good saves before going off injured, while at the other end Tim Howard denied Markus Rosenberg and Shane Long.
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 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.
Mirallas showed great skill to take a touch and then shoot from the edge of the area after a good run by Coleman but Foster tipped over.
Everton began the second half determined to make their superiority count but Nikica Jelavic was unable to turn in a low cross from Leighton Baines, who was also subject of a failed bid by Manchester United this week along with Fellaini.
Barkley struck a sweet left-foot shot from the edge of the area but Foster saved low to his right and Fellaini miscued the rebound.
Everton maintained the tempo with Mirallas shooting straight at Foster after turning inside the area.
The hosts began to grow frustrated and were almost caught out on the counter-attack when Morrison connected well from distance but Howard made his first significant save.
Everton almost paid the price for some sloppiness 15 minutes from time when substitute Steven Naismith was robbed by Morrison, who quickly set up Long, but again Howard saved well.
West Brom boss Clarke was forced to change his goalkeeper 12 minutes from time as Foster limped off.
Luke Daniels took his place and was beaten all ends up as Fellaini brilliantly took a Phil Jagielka cross on his chest and dabbed past, only for the ball to rebound off the post.
West Brom had a scare with 90 minutes on the clock when Coleman’s cross dipped over Daniels and struck the bar, while Everton also appealed unsuccessfully for a penalty after a Barkley shot struck Billy Jones.