Fulham's Danny Murphy and Clint Dempsey ran the show at The Hawthorns but neither side could break the deadlock.
The visitors thought they had scored when Pajtim Kasami pounced after Ben Foster had spilled Dempsey's shot, but the Swiss was ruled offside.
Peter Odemwingie spurned West Brom's best chance, shooting tamely at Mark Schwarzer when clean through.
Bryan Ruiz came within inches of a late Fulham winner but stabbed wide after another Foster fumble.
Both teams have endured a disappointing start to the Premier League campaign.
The visitors, still without a league win this season, dominated the first period with skipper Murphy dictating play, but Albion posed more of a threat in the second half.
Odemwingie represented the biggest danger to the Londoners but Fulham held firm, with centre-back Brede Hangeland marshalling his defence well.
West Brom were the first to threaten, skipper Chris Brunt getting on the end of a Nicky Shorey cross but his powerful header was directed straight at Schwarzer.
Then Youssouf Mulumbu dispossessed Kasami 20 yards out and his low drive forced a sprawling save from the Australian, diving to his right.
The Baggies reverted to 4-4-2 formation from a 4-5-1 set-up for the second period, with Odemwingie pushing further forward to support Shane Long.
The woodwork came to West Brom's rescue when defender Jonas Olsson inadvertently deflected a volley by Dempsey against his own bar.
Then Moussa Dembele found himself unmarked 15 yards out but could not keep his shot down.
Albion appealed in vain for a penalty after a challenge on Long by Chris Baird, but the Irish forward appeared to make the most of what little contact there had been.
Odemwingie seemed sure to put the home side in front after 64 minutes but after being played in behind the static Fulham defence by Mulumbu's inch-perfect pass, he could muster only a weak shot and Schwarzer saved comfortably.
Fulham came close to securing all three points with 12 minutes remaining when John Arne Riise's stinging free-kick was saved by Foster, but substitute Ruiz, a £10.6m summer arrival from FC Twente, watched in vain as his toe-poked effort trickled wide.
Replays showed that Foster got the slightest of touches as he dived at the Costa Rican's feet but referee Stuart Attwell gave a goal kick.
Brunt nearly earned Albion victory in injury time when his low cross-shot hit the post, but in the end both sides seemed content with a point.
Fulham manager Martin Jol:
"I know West Brom. I played here. It looked like an away game for them. It was good. We had all the possession, we could play around them in midfield.
"They changed their system, playing three up instead of two strikers, and we could play in the gaps. The only thing is you need to finish them off. We had three or four possibilities. I think we looked the better team.
"We had the ball against the bar and post, we had a few shots, we can probably do better in the future. I feel if you play the way we do, with four or five draws, it is a bit disappointing and I think we are in a false position."
West Brom manager Roy Hodgson:
"In the first half we didn't play at all well. I take some responsibility for that because we tampered with our way of playing and the system.
"It didn't work particularly well and I didn't even recognise some of the players in terms of the number of times we gave the ball away.
"In the second half we pushed Peter Odemwingie more central again, Chris Brunt out wide, and went back to a more classic 4-4-2 system and dominated the game."