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.
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.
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.
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.
