Sigurdsson, who had been a doubt because of a foot injury, latched onto a long ball into the penalty area with just three minutes left to send the Championship side into the fifth round.
The result was no more than the hosts deserved for their efforts, as Brian Laws' side were made to pay for a lack of any real ambition.
There was an early opening for Reading when Bryn Gunnarsson fed Sigurdsson on the left.
His low centre was turned goalwards by Grzegorz Rasiak, but the effort lacked any power to trouble Brian Jensen.
The Polish striker had another chance on 11 minutes when he got up to meet Jobi McAnuff's deep centre from the right, but headed over.
Burnley beat Reading in last season's play-off semi-finals, which marked the end of Steve Coppell's reign.
Since then the Royals - relegated from the Premier League in 2008 - have been on a downward spiral.
Nevertheless, Brian McDermott, in temporary charge following the sacking of Brendan Rodgers, has installed some fight in the side, and they continued to take the game to Burnley.
Sigurdsson - whose stoppage-time penalty at Anfield made it eight for the season - tested Jensen when he created space some 20-yards out.
The homes fans, what there were of them in a below-capacity attendance of less than 13,000, called for a penalty when Rasiak went down as he and Tyrone Mears challenged for a cross into the six-yard box, but referee Andy D'Urso was not impressed.
Rasiak was then again played in down the left side of the penalty area, but once more he failed to make a decent connection.
As the first half drew to a close, former Manchester United trainee Chris Eagles tried his luck with an ambitious over-head kick, which drifted harmlessly wide.
Burnley made a change for the second half, as Martin Paterson replaced Steve Thompson.
Laws clearly wanted to inject fresh life into his side and there was at least a bit more impetus from the visitors following the restart.
However, delivery from promising positions was poor.
Burnley keeper Jensen was forced into action when Simon Church's shot-on-the-turn looked to be heading towards the top-left corner.
Reading then had a free-kick right on the edge of the penalty area after Church was brought down by Michael Duff.
Sigurdsson, though, chipped his effort over the bar.
At the other end, Wade Elliot teed up captain Graham Alexander 20 yards out, and his low shot flew just wide.
As the hour mark approached, Burnley enjoyed their best spell.
Joey Gudjonsson arrived at the near post to meet Eagles' ball in, with his shot blocked on the line.
With 25 minutes left, Reading made a change, as Shane Long, who hit the winner in extra-time at Anfield, replaced Church.
Reading were again screaming for a penalty as McAnuff went down under a challenge from Gudjonsson, but despite there looking to be contact, D'Urso again waved play on.
To add to the frustrations of the home fans, Sigurdsson then headed wide from six yards.
Kevin McDonald forced Adam Federici into a full-stretch save as his low shot from 20 yards looked to be creeping into the bottom corner, before the Reading goalkeeper then clawed away David Edgar's header and Paterson blasted the rebound over.
Jensen tipped over a header from Mills and captain Ivar Ingimarsson blazed his effort wide as the hosts pressed.
Reading were rewarded with just three minutes left as Andy Griffin's deep ball found its way through to the six-yard box, where Sigurdsson guided his shot into the bottom corner to leave the Wembley dream alive for the home fans.