Shelton Martis hit the bar at Upton Park and Jonathan Greening missed the clearest chance of a lifeless clash - and Tony Mowbray's men will consider the stalemate as two points dropped.
The draw moved them six points from safety but it could so easily have been a first top-flight win in London after 10 failed attempts.
Greening rued a missed chance to take a rare away win.
He told Setanta Sports: "I think I probably should have put my chance away and got us all three points.
"We knew it was going to be a difficult game, Gianfranco Zola has got them playing well, but I thought we did all right tonight.
"We've been working hard all season. It's difficult coming up from the Championship but the boys have worked hard and we'll give it a good go until the end of the season."
Baggies boss Mowbray said: "I don't think it was an improved performance - the defence was better but the performance was what we get most weeks.
"We missed some chances and it's a very disappointed changing room there. But it's something to build on."
THE PREMIERSHIP THIS WEEKEND
Saturday 21st March 2009
Blackburn v West Ham 15.00
Fulham v Man Utd 15.00
Newcastle v Arsenal 17.30
Portsmouth v Everton 12.45
Stoke City v Middlesbrough 15.00
Tottenham v Chelsea 15.00
West Brom v Bolton 15.00
Sunday 22nd March 2009
Liverpool v Aston Villa 16.00
Man City v Sunderland 15.00
Wigan Athletic v Hull City 13.30