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Matej Vydra struck with four minutes left to deny new Fulham manager Felix Magath victory in his first match in charge.
It had looked as though Magath was going to be indebted to a familiar face in Ashkan Dejagah, who won the Bundesliga title with the German manager in 2008-09 in his spell at Wolfsburg and put Fulham ahead after 28 minutes.
But the Barclays Premier League's bottom club were prevented from claiming their first win in seven League matches when Vydra's shot squirmed under goalkeeper Maarten Stekelenburg in the closing stages.
The result leaves Fulham four points adrift of safety at the foot of the division, while West Bromwich Albion are a point above the drop zone in 16th, although head coach Pepe Mel is still waiting for his first win since taking over last month.
January signing Konstantinos Mitroglou was named on the bench for Fulham, who started brightly and could have gone ahead when two other new recruits combined as Johnny Heitinga headed over from Lewis Holtby's left-wing cross.
Fulham continued to dictate play without creating too many chances, although Steve Sidwell did strike over from point-blank range after 21 minutes following good work down the right from Hugo Rodallega.
But for all their pressure, the visitors were thankful to Stekelenburg for keeping them level, the Netherlands international standing firm to deny Thievy Bifouma after the Frenchman had combined brilliantly with compatriot Morgan Amalfitano on the counter-attack.
Magath's men were undeterred, though, and deservedly took the lead just before the half-hour mark when Dejagah collected Kieran Richardson's delivery before powering a right-footed finish beyond Ben Foster.
West Brom struggled to mount a response and would have fallen further behind had Foster not pulled off a fine diving save to keep out Rodallega's long-range drive.
And the hosts' cause was further damaged when defender Billy Jones was withdrawn through injury and replaced by Victor Anichebe.
Mel's men appeared reinvigorated after half-time, though, with Stekelenburg forced into another impressive save to prevent Chris Brunt from levelling matters.
The Hawthorns outfit were dominant throughout the second period but Steklenburg's inspired performance continued as he prevented James Morrison from equalising after 66 minutes with a vicious effort from the edge of the area.
It looked as though West Brom's effort were going to be in vain, but the Midlands club salvaged a point in dramatic fashion when substitute Vydra's effort crept under Stekelenburg.
Stekelenburg produced a brilliant save to keep out an injury-time strike from Brunt, with Anichebe firing wide on the rebound.
West Brom manager Pepe Mel: "One point is one point but I think today we were too open in the first half and the second half was good. We needed two points more.
"We need to win for our mentality. The players need to win.
"I believe in the next match against Manchester United we can win. There is a nervousness in the changing room."
Fulham manager Felix Magath: "We were so near to a victory, we are disappointed but in the whole game, 1-1 was correct. It was a fair result.
"We have had only a few days together. You cannot change many things.
"I crossed the fingers but that is not enough. We have to play 90 minutes and try to make no faults. We made too many faults in the second half.
"We lost the ball too early and came under pressure. If you get under pressure, one mistake will cost you a goal.
"We were away and I think it was not very good but a point is good."
|21 Apr 2014||v Man City (A)|
|05 Apr 2014||v Norwich (A)|
|04 May 2014||v Arsenal (A)|
|29 Mar 2014||v Cardiff (H)|
|12 Apr 2014||v Spurs (H)|
|26 Apr 2014||v West Ham (H)|