Reading were trailing 2-1 in the third minute of added time when Brian Howard drew them level to force extra time.
Iceland Under-21 international Sigurdsson then scored a stunning winner from 25 yards in the 95th minute to set up a home tie in the last eight against Aston Villa.
No-one in the Reading camp would have been more relieved by the result than goalkeeper Adam Federici.
He looked to have handed Albion victory when he let Robert Koren's 47th minute shot slip through his fingers and over the line.
Koren, who had also been on target in the first encounter between the sides, had given Albion a sixth-minute lead.
Jimmy Kebe, whose struck in the first game after just nine seconds, dragged Reading level in the ninth minute before Federici's second-half gift looked to have ended their cup adventure, only for Howard and Sigurdsson to seal the most dramatic of comebacks.
Albion made a frantic start to the game and were rewarded when Koren capped a spell of intense pressure.
Even before Koren struck, the Slovenia international and Roman Bednar had been denied by Federici while Abdoulaye Meite headed just over the bar.
But Federici couldn't deny Koren a second time despite his best efforts.
Federici had saved a low drive from Bednar and then Ryan Bertrand cleared the follow-up from Luke Moore off the line only to see the loose ball fly to Koren, who stabbed the ball into an empty net from six yards.
Any hopes Albion had that one goal would be enough to book their place in the last eight disappeared just three minutes later.
Like Koren, Kebe couldn't miss from close range after he pounced when Scott Carson could only palm out a shot from Sigurdsson following a cross from Simon Church.
Failing to hold on to their advantage, rocked Albion back on their heels and they were fortunate not to fall behind shortly after when Matt Mills wasted a golden chance by heading wide and Jobi McAnuff was denied by Carson.
Albion reserved their best play for the opening stages of each half and they started the second period at a fast pace but could not have envisaged the helping hand they would get from Federici.
Youssouf Mulumbu picked up a throw-in from Gianni Zuiverloon and fed Koren on the edge of the area.
Koren's shot didn't carry that much power but inexplicably Federici let it slip from his grasp at the near post.
Reading did try and launch an immediate response and Sigurdsson hit the crossbar from 25 yards less than 60 seconds later with Carson saving the follow-up from Church.
Sigurdsson was then denied by Carson in the 70th minute and ultimately Reading had a helping hand of their own to draw level for a second time.
Zuiverloon missed his clearance and, when the ball broke to Kebe, he fed Howard who ended a surging run into the area by blasting past Carson from 12 yards before Sigurdsson's 10th goal of the season sealed Reading's success.