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.
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.
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."