Sidwell's first-half strike put the hosts ahead before Dimitar Berbatov (twice) and Hugo Rodallega had efforts disallowed, while Bryan Ruiz was also denied by the crossbar.
And 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 leveller 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.
With record signing Stephane Sessegnon absent due to work permit complications, Victor Anichebe led the line for West Brom alongside Nicolas Anelka.
Winless and scoreless in their opening three league games, West Brom fell behind after 21 minutes as Sidwell put the hosts 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 Steve Clarke's men almost fell further behind after 32 minutes as the home side threatened to double their lead.
Promising build-up play saw Sidwell guide the ball into Dimitar Berbatov's path across the penalty area, and the former Bulgaria international slotted the ball home with ease from a few yards out.
However, the officials rightly ruled the goal out because Berbatov was stood 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 former Tottenham and Manchester United striker found himself through on goal and despite finding the back of the net, the strike was again disallowed.
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 equaliser, 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 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 their 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.
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