The Cottagers had three goals disallowed for offside but could not extend their lead, and were eventually made to pay by a stoppage-time header from the Albion center back.
Steve Sidwell's first-half strike put the home side ahead before Dimitar Berbatov, twice, and Hugo Rodallega had efforts disallowed, while Bryan Ruiz was also denied by the crossbar.
McAuley made Fulham pay in dramatic fashion as he ended West Brom's wait for a first Premier League goal of the season with a last-gasp leveler with a late header.
Former West Brom player Martin Jol made three changes to his Fulham side, handing starts to Philippe Senderos, Pajtim Kasami and Kieran Richardson. Aaron Hughes and Darren Bent were both unavailable through injury.
|By Liam Twomey
The final whistle was greeted with boos and very audible shouts of 'Jol out' at Craven Cottage, after Gareth McAuley's late header denied the hosts what would have been an important win.
Fulham had lost their last five home matches before today, and throughout the afternoon there was very audible frustration at the home side's failure to kill the game off against a pretty poor looking West Brom side.
Jol's worries will only be multiplied by what appeared to be an Achilles injury to Bryan Ruiz only moments before McAuley's equalizer. The Costa Rican was stretchered off in obvious agony, and it could be quite a while before we see him on a football pitch again.
Winless and scoreless in its opening three league games, West Brom fell behind after 21 minutes as Sidwell put the home side ahead.
A threatening Alexander Kacaniklic corner saw Senderos challenge for possession before Sidwell struck the ball home past goalkeeper Boaz Myhill.
Fulham's confidence visibly lifted after that opening goal, and Clarke's men almost fell further behind after 32 minutes as the home side threatened to double its lead.
Promising build-up play saw Sidwell guide the ball into Berbatov's path across the penalty area, and the former Bulgaria international slotted home with ease from a few yards out. However, the officials rightly ruled the goal out because Berbatov was in an offside position.
Seven minutes into the second half, Berbatov dramatically had another goal rightly ruled out for offside. Cleverly combining with Kasami in the attacking third, the 32-year-old found himself through on goal and despite finding the back of the net, the strike was again waved off.
Anichebe, a deadline-day signing from Everton, should have scored moments later but his weak shot was fired directly at Fulham goalkeeper David Stockdale.
In a bid to find an equalizer, Clarke introduced Scott Sinclair and James Morrison into proceedings, while Fulham's Fernando Amorebieta made history by coming off the bench to become the first Venezuelan to play in the Premier League.
A searching ball from Ruiz after 65 minutes almost saw Berbatov score Fulham's second, but his spectacular, acrobatic volley soared just over the bar.
West Brom then had genuine penalty claims as Anelka was brought down under pressure from Senderos, but referee Lee Probert controversially waved away the protests.
Ruiz calmly guided the ball goalwards after 84 minutes but his strike cannoned back off the bar, before substitute Rodallega had Fulham's third goal of the afternoon disallowed.
Having upped the attacking intent in the second half, McAuley then secured a deserved point for West Brom with a dramatic injury-time headed goal following good play from Morrison to make Fulham pay for its profligacy.