On a day that saw Wales' rugby team seal a Six Nations Grand Slam, Brendan Rodgers’s side - off the back of a shock 1-0 win at home to Manchester City - were barely threatened by the Whites, whose individual errors Sigurdsson exploited to take his tally to five goals in eight matches since joining on loan from Hoffenheim.
Joe Allen added a late third, again after terrible play by the hosts, as Fulham lost at home for the first time since the 5-0 thrashing by Manchester United late last year.
Ex-Reading midfielder Sigurdsson, who strangely fell out of favour in Germany despite impressing in his first season in the Bundesliga, netted a first-half diving header and a second-half drive after starting counter-attacks handed to him by sloppy play from Pavel Pogrebnyak and Mahamadou Diarra respectively.
Allen’s late goal confirmed Fulham's hapless performance, coming after a catalogue of errors, Philippe Senderos twice culpable while Mark Schwarzer should have saved his bobbling shot.
The Swans leapfrog Fulham to eighth, one point short of the magic 40 and all-but assured top-flight football next season.
Martin Jol paid the price for an eccentric team selection, resting Danny Murphy and Damien Duff and deciding not to recall Aaron Hughes after his return from injury: stand-ins Diarra, Brian Ruiz and Senderos were woeful, the former and latter responsible for two goals and the Costa Rican incapable of completing a pass.
After a bright start by both sides, the hosts seemed to slack off, unable to retain possession thanks to pressure from their guests but making some sloppy errors, from one of which they conceded.
The opening few minutes saw Clint Dempsey’s solo run and shot force Michael Vorm to tip over, with the Dutchman doing well immediately afterwards as Moussa Dembele fired a low effort.
But it was in contrast to the rest of the half where the Whites, normally confident in possession at home, were harried by energetic Swansea, the hosts missing Murphy’s influence and not helped by Ruiz, the winger blowing hot and cold at the best of times and very much chilly this time out as he constantly gave the ball away.
The visitors first real opportunity came just before the half-hour mark when Sigurdsson drilled a low shot wide of the far post, his influence increasing as the half progressed.
Swansea’s pressing approach was also forcing errors, one of which was a poor touch and clearance by Schwarzer, closed down by Danny Graham: the Australian goalkeeper made amends with a fine save from Scott Sinclair.
So the Whites had been adequately warned when, 38 minutes in, Pogrebnyak blotted his previously excellent Fulham copybook with a misplaced pass to Sigurdsson: he quickly offloaded the ball for Wayne Routledge to send a deep cross, continuing his run to meet Sinclair’s cut-back with a diving header past Schwarzer.
The visitors deserved it by default, as Fulham were asking to be punished: the setback provoked a response from the home side, but they were unable to find a good final ball or finish to test Vorm and his defence.
Fulham were brighter out of the blocks in the second half, although they failed to truly hurt Swansea as Ruiz continued to misplace his passes, while Pogrebnyak - having scored five goals in his first three matches for the club - was let down by his touch on more than one occasion.
Ruiz was brought off for Duff, who immediately went on a direct run but fired well over. Indeed, the Irishman appeared a better option, later cutting in from the right but seeing his drive blocked by Neil Taylor.
But the hosts still failed to open Swansea up, Vorm still untroubled as Jol’s side toiled in the final third, Andy Johnson mostly anonymous before he was hauled off for Kerim Frei.
Swansea appeared to be playing within themselves, waiting for an error. And it soon came: midway through the half Diarra and Dembele were both stranded upfield when a high ball bounced into their half, Sigurdsson quickly releasing Routledge - unlike the first goal, his cross found the Icelandic immediately for a crisp low finish inside the bottom right.
Diarra was the main culprit as his role is primarily to protect the defence, yet he was loitering in the opposition half: the Malian free-transfer signing soon came off for Murphy, but it was far too late.
Again Fulham tried to respond, pushing hard but only half-testing Vorm when Dempsey drilled a low shot wide.
And their defensive horrorshow continued when, after he gave the ball away to Sinclair, Senderos fell over as Allen cut inside, allowing the Wales midfielder to release a shot that Schwarzer allowed to bounce in off his hand.
After that comedy of errors Craven Cottage started to empty - England’s upcoming rugby match against Ireland starting immediately afterwards - but all they missed were a Dempsey shot off the bar and a Pogrebnyak effort that flew over.
There was the unusual sight of Swansea making a triple-substitution in the final minute of stoppage time as they won a mid-table six-pointer at a canter, the away support singing in double celebration - their national rugby team's Grand Slam had been confirmed late in the second half.