Chances were few and far between in the early stages of the game, but it was the hosts who opened the scoring midway through the first half when Steve Sidwell fired past Boaz Myhill on the turn after Alexander Kacaniklic's corner fell into his path.
Dimitar Berbatov found himself denied by the offside flag twice, with a marginally close call ruling out his second attempt before he almost doubled Fulham's lead with an acrobatic effort after a neat ball from Bryan Ruiz.
The introductions of Scott Sinclair, James Morrison and Saido Berahino gave West Brom an added impetus and they eventually levelled when Gareth McAuley guided Chris Brunt's last-minute corner past a helpless Stockdale.
Fulham had looked like ending a five-match winless streak at Craven Cottage following Sidwell's first-half strike but now the run is six since the April 1 defeat of QPR.
The Baggies had been toiling up until the equaliser, but returned to the West Midlands content, while Martin Jol's hosts and their fans were left frustrated, a vocal minority even calling at the final whistle for the Dutchman to be relieved of his duties.
West Brom had not found the net since the final day of last season, but prospects of ending the drought appeared promising - on paper at least - as Steve Clarke deployed a heavyweight strike-force in Anichebe and Nicolas Anelka.
Anichebe and on-loan Marseille midfielder Morgan Amalfitano made their Baggies bows, but Stephane Sessegnon, another deadline day signing, missed out due to a delay in obtaining his work permit.
Amalfitano was part of a visiting midfield which stifled Fulham's efforts in the final third until Ruiz flashed an early effort wide after Pajtim Kasami's cross was only half cleared.
Kacaniklic's set-piece found Philippe Senderos, who held off McAuley to nod down for Sidwell to hit a shot on the turn between Youssouf Mulumbu, who was on the post, and goalkeeper Myhill.
Berbatov was clearly offside when he had a strike ruled out after 32 minutes before Brunt tried an audacious shot from 40 yards to revive Albion, but the midfielder shot wide of David Stockdale's goal.
Berbatov was adjudged to be offside once more when tucking in a Kasami pass. Had the marginal call gone in the Bulgarian's favour then Fulham would have been cruising but as it was the visitors were still in the contest.
The Baggies should have been level when Anichebe latched onto Brunt's pass and lost Brede Hangeland, but the former Everton forward's shot lacked conviction as Stockdale blocked with his legs.
Berbatov had a legitimate opportunity when making an arching run to meet Ruiz's superb pass, but Myhill was alert to save.
The same combination went close when Ruiz drifted in field and lofted a pass over which Berbatov met acrobatically, directing the ball just wide.
The visitors needed a lift and it almost came when substitute Morrison's pass was brought down and held up by Anichebe.
Ruiz hit the bar six minutes from time and substitute Hugo Rodallega netted the rebound, only for the strike to be ruled out for offside.
Fulham, who saw Ruiz depart on a stretcher after an awkward fall, were made to pay.
The Baggies won a corner and Brunt's delivery was met by McAuley, whose header flew into the net.