Somen Tchoyi snatched a late equaliser for West Brom just five minutes after coming off the bench.
Fulham looked to be cruising to victory after Clint Dempsey opened the scoring from close range midway through the second half.
But Tchoyi ran on to James Morrison's flick to equalise with a cross-shot for his first goal of the season.
Albion have lost only one of their last five Premier League away games and are eight points clear of the bottom three.
Fulham manager Martin Jol started without a recognised striker after selling Bobby Zamora to QPR in a £4 million deal , with Russian forward Pavel Pogrebnyak ineligible after joining from Stuttgart and Andy Johnson and Moussa Dembele absent through injury.
Did you know?
Other than ten-goal Clint Dempsey, no player in Fulham's starting line-up has managed more than two Premier League goals this season.
And, despite Dempsey opening the scoring with his 11th goal in 13 games in all competitions, this performance did little to dispel concerns that they may struggle for goals in the second-half of the season.
Other than that, and an effort that Dempsey had ruled out for offside in the first half, the Cottagers rarely looked like stretching a West Brom side that has picked up 18 of their 26 points away from home.
Fulham controlled possession from the start but struggled to create decent openings, with Steve Sidwell missing from distance on three occasions in the first half.
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The hosts did have the ball in the net after quick passes from Damien Duff and Danny Murphy played Dempsey but the American was offside.
Peter Odemwingie, who was a target for Fulham in the January transfer window, sliced well wide with his first chance after a good move involving Graham Dorrans and James Morrison.
The Nigerian did lay on a decent opening for Marc-Antoine Fortune, when he stepped inside Stephen Kelly and crossed, but the West Brom striker was denied by a low save from David Stockdale.
Fulham's only other chance of the first half came when Ben Foster rushed out and missed Danny Murphy's long ball forward but Bryan Ruiz's header dropped the wrong side of the post.
Simon Cox had a curling effort from the right flank held by Stockdale at the start of the second half but Fulham were still struggling to find any penetration.
West Brom showed another example of their counter-attacking potential when Odemwingie found space down the right and crossed for Fortune, who was denied by a low save from Stockdale.
Bryan Ruiz showed some class when he slipped past Gareth McAuley's challenge but could only prod tamely straight at Foster.
John Arne Riise showed Fulham's growing frustration by trying his luck from almost 30 yards and Duff also clipped a cross just wide of the far post after a run that beat three players.
But with 22 minutes remaining a disguised pass from Ruiz put Dempsey through a gap and he thumped past Foster.
West Brom had shown no sign of a response but Tchoyi latched on to Morrison's pass, held off Brede Hangeland and hit a shot with enough power to beat Stockdale.
Dempsey passed up a chance to score a second when he met Murphy's corner unmarked at the far post but headed high and wide of goal to ensure a happy return to Craven Cottage for Roy Hodgson.
Fulham manager Martin Jol
"To concede eight or nine minutes before the end, and with it being a bit unlucky as well, you are disappointed but they were difficult circumstances because Andy Johnson pulled out, Bobby (Zamora) has gone to QPR and Moussa Dembele couldn't play.
"Clint scores goals from the left and up-front. It's extraordinary to score goals the way he does.
"He is normally on the left but today he played up-front and you know there are goals in him so I was pleased."
West Brom manager Roy Hodgson:
"When they scored I was concerned because we'd done well up to that point and looked more likely. We were worth a point and when they scored you fear the worst but we got back in it but the last five minutes or so could have been calmer.
"The crowd were fantastic and everyone at the club was nice to me which was great. I wouldn't have been as happy had we lost but getting the point and the reception [I did] makes it a wonderful evening.
"The squad looks as though it will get stronger and that will be necessary in the last 15 games because we know it will be a dog-fight between a lot of teams trying to stay away from that wretched relegation zone that strikes fear into so many hearts."
Live Text Commentary
Full Time The referee blows for the end of the match.90:00 +3:46
John Arne Riise takes a shot. Youssouf Mulumbu gets a block in.90:00 +2:51
Bryan Ruiz fouled by Youssouf Mulumbu, the ref awards a free kick. Bryan Ruiz restarts play with the free kick.87:59 Substitution
Substitution James Morrison leaves the field to be replaced by Steven Reid.87:27
Danny Murphy takes the inswinging corner, Clint Dempsey has a header from deep inside the six-yard box and clears the crossbar.87:12
Danny Murphy takes a inswinging corner from the left by-line to the near post, Gabriel Tamas makes a clearance.86:46
Corner taken by Danny Murphy played to the near post, clearance made by Gabriel Tamas.85:28
James Morrison gives away a free kick for an unfair challenge on Bryan Ruiz. Free kick crossed by Danny Murphy.81:49
Assist by James Morrison.81:49 Goal scored
Goal! - Somen Tchoyi - Fulham 1 - 1 West Brom Somen Tchoyi fires in a goal from inside the six-yard box to the bottom left corner of the goal. Fulham 1-1 West Brom.80:50
Damien Duff delivers the ball.76:25 Substitution
Substitution Somen Tchoyi is brought on as a substitute for Simon Cox.75:31
Graham Dorrans takes the inswinging corner, John Arne Riise manages to make a clearance.75:19
Gabriel Tamas produces a cross, Brede Hangeland manages to make a clearance.74:57
Inswinging corner taken right-footed by Graham Dorrans from the left by-line, David Stockdale makes a save.74:52
A cross is delivered by Peter Odemwingie, Damien Duff gets a block in.74:32
The ball is delivered by Nicky Shorey, clearance by Brede Hangeland.73:45
The ball is crossed by Nicky Shorey, blocked by Stephen Kelly. Corner from the left by-line taken by Graham Dorrans, clearance made by Brede Hangeland.72:38
Damien Duff delivers the ball, save made by Ben Foster.71:47
Free kick awarded for an unfair challenge on Bryan Ruiz by Jonas Olsson. Danny Murphy takes the free kick.
The assist for the goal came from Bryan Ruiz.68:42 Goal scored
Goal! - Clint Dempsey - Fulham 1 - 0 West Brom Clint Dempsey finds the back of the net with a goal from inside the box to the bottom left corner of the goal. Fulham 1-0 West Brom.68:26
Centre by Stephen Kelly, clearance made by Jonas Olsson.67:54
A cross is delivered by Stephen Kelly.64:20
Effort from inside the area by Peter Odemwingie misses to the right of the goal.62:51
Peter Odemwingie takes a shot. Blocked by Steve Sidwell.61:35
The ball is swung over by Damien Duff.60:11
Shot by Peter Odemwingie from outside the box goes over the net.59:12
Effort from outside the penalty area by John Arne Riise goes wide left of the target.59:00
The ball is crossed by Damien Duff, clearance made by Gareth McAuley.58:14
Bryan Ruiz takes a shot. Save by Ben Foster.57:41
Graham Dorrans takes a shot. Save made by David Stockdale.56:18
Inswinging corner taken right-footed by Graham Dorrans played to the near post, clearance made by Simon Davies.55:12
Free kick awarded for an unfair challenge on Clint Dempsey by Gareth McAuley. Free kick taken by Danny Murphy.53:06
Inswinging corner taken by Damien Duff from the right by-line, clearance by Marc-Antoine Fortune.50:36
Free kick crossed by Graham Dorrans, clearance made by Brede Hangeland.50:36 Booking
Booking The referee cautions Danny Murphy for unsporting behaviour.50:20
Danny Murphy gives away a free kick for an unfair challenge on Graham Dorrans.49:44
Centre by Graham Dorrans, David Stockdale makes a save.47:51
Simon Cox sends in a cross, Philippe Senderos manages to make a clearance.47:01
Simon Cox takes a shot. Save made by David Stockdale.46:01
Unfair challenge on Marc-Antoine Fortune by Brede Hangeland results in a free kick. Peter Odemwingie has an effort direct from the free kick.45:01
The second half kicks off.45:00 +1:14 Half time
Half Time The players leave the pitch at half time.45:00 +0:14
Danny Murphy sends in a cross, clearance by Jonas Olsson.44:01
The assistant referee flags for offside against Peter Odemwingie. Indirect free kick taken by Brede Hangeland.43:36
Clint Dempsey takes a shot. Save made by Ben Foster.42:12
Free kick awarded for an unfair challenge on Youssouf Mulumbu by Steve Sidwell. Gabriel Tamas takes the direct free kick.41:32
Marc-Antoine Fortune concedes a free kick for a foul on Danny Murphy. Danny Murphy takes the direct free kick.41:14
The ball is crossed by Steve Sidwell.40:57
John Arne Riise produces a cross, clearance made by Jonas Olsson.40:46
Danny Murphy produces a cross, Gabriel Tamas manages to make a clearance.40:29
The ball is swung over by John Arne Riise, clearance made by Nicky Shorey.39:45
James Morrison takes a shot. Philippe Senderos gets a block in.38:10
Danny Murphy gives away a free kick for an unfair challenge on Youssouf Mulumbu. Free kick taken by Gabriel Tamas.37:41
Danny Murphy delivers the ball, Gabriel Tamas manages to make a clearance.36:47
Unfair challenge on Youssouf Mulumbu by Steve Sidwell results in a free kick. Gabriel Tamas takes the free kick.35:11
Damien Duff takes a shot. Jonas Olsson gets a block in.33:37
Clint Dempsey has an effort at goal from long range that misses to the right of the target.31:25
Damien Duff sends in a cross.30:55
Youssouf Mulumbu concedes a free kick for a foul on Bryan Ruiz. Direct free kick taken by Danny Murphy.29:37
Shot on goal by Marc-Antoine Fortune from just inside the area goes over the target.26:36
Inswinging corner taken by Simon Davies from the left by-line.26:01
Danny Murphy produces a cross, Bryan Ruiz takes a shot. Save by Ben Foster.24:52
Nicky Shorey takes a inswinging corner to the near post, Brede Hangeland manages to make a clearance.24:37
Gabriel Tamas takes a shot. Save made by David Stockdale.23:40
Danny Murphy takes a shot. Blocked by Graham Dorrans. Shot from just outside the box by Steve Sidwell misses to the right of the goal.22:20
Graham Dorrans takes the inswinging corner, Gabriel Tamas takes a shot. Save by David Stockdale.22:00
Graham Dorrans takes a shot. Blocked by Brede Hangeland.20:54
Corner taken by Danny Murphy, save by Ben Foster.18:58
Effort on goal by Steve Sidwell from outside the area goes harmlessly over the bar.18:30
A cross is delivered by Damien Duff, Jonas Olsson manages to make a clearance.16:16
A cross is delivered by Damien Duff, Gareth McAuley manages to make a clearance.13:37
Damien Duff takes the inswinging corner, clearance by Gareth McAuley.13:01
Inswinging corner taken left-footed by Damien Duff, clearance made by Marc-Antoine Fortune.12:00
Effort from deep inside the area by Peter Odemwingie misses to the right of the goal.11:15
Marc-Antoine Fortune is caught offside. Indirect free kick taken by Philippe Senderos.10:46
Free kick awarded for an unfair challenge on Marc-Antoine Fortune by Steve Sidwell. James Morrison takes the direct free kick.10:13
The assistant referee signals for offside against Clint Dempsey. Free kick taken by Ben Foster.9:35
Steve Sidwell crosses the ball, clearance by Gabriel Tamas.8:04
The referee penalises Simon Cox for handball. John Arne Riise restarts play with the free kick.7:46
Centre by Simon Cox, clearance by Damien Duff.7:08
Inswinging corner taken right-footed by Graham Dorrans from the left by-line, David Stockdale makes a save.6:40
James Morrison takes a shot. David Stockdale makes a comfortable save.4:49
Simon Cox takes a shot. Save by David Stockdale.3:35
Unfair challenge on Stephen Kelly by Marc-Antoine Fortune results in a free kick. David Stockdale takes the direct free kick.1:59
Steve Sidwell has an effort at goal from outside the penalty area missing to the wide left of the goal.0:00
The referee gets the match started.
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