Second-half goals from Leighton Baines and Nikica Jelavic saw Everton leapfrog Swansea in the Barclays Premier League table and become just the third away side to win at the Liberty Stadium this season.
Baines' dusted himself down after being fouled by Ashley Williams to curl a lovely free-kick beyond Michel Vorm just before the hour.
Jelavic sealed the win by tapping in his second goal for the Toffees with 14 minutes remaining.
Much had been made of Swansea's potential chances of European qualification following a run of three-straight wins and they enjoyed the better of the first half but could not make it pay against a disciplined Everton rearguard.
After a low-key opening the first chance fell Swansea's way in the seventh minute after Sigurdsson's pass had played Danny Graham in down the right channel, but the striker could not get hold of his shot and it was easily held by Tim Howard.
As with most visitors to the Liberty Stadium this season, Everton had to cope without the ball for long periods and Wayne Routledge blazed a presentable chance over after Joe Allen's cushioned lay-off, while Phil Jagielka bravely cleared a dangerous Sigurdsson cut-back seconds later.
Everton's first effort on goal came after 27 minutes as Jelavic headed wide from Baines' free-kick with Leon Osman poised in a better position behind him.
But Swansea were soon back in control and Jagielka was again Everton's saviour as he blocked Sigurdsson's shot after the Iceland international had been picked out by a lovely outside-of-the-foot pass from Routledge.
But the pattern of the game altered after the break as the likes of Steven Pienaar and Osman, peripheral figures up to that point, began to feature more prominently.
They could have led moments after the restart but Jelavic could not keep his effort down as he stretched for a loose ball after Osman had charged down a Williams clearance.
Swansea keeper Vorm had been a virtual spectator but he was alert enough to beat away a fierce Darron Gibson drive, with Tim Cahill failing to connect as Jelavic helped the rebound back across goal.
Graham then headed over from Routledge's cross with Howard stranded as the contest finally showed signs of opening up, while Pienaar shot narrowly wide at the other end as the introduction of substitute Marouane Fellaini began to influence play after replacing Cahill.
And, having soaked up so much first-half pressure, the Toffees grabbed the lead just before the hour as Baines stepped up to curl a left-footed free-kick into the right corner of the net after Williams had brought the England international down on the edge of the box.
Swansea boss Brendan Rodgers made a triple substitution as he sought to change the flow of the game as Graham, Scott Sinclair and Routledge made way for Luke Moore, Josh McEachran and Leroy Lita.
But as the hosts pressed forward they left themselves vulnerable to the counter and, after Pienaar had curled an effort wide, the South African teed up Jelavic who fired horribly off target with just Vorm to beat.
The former Rangers' striker then missed another fine chance as he failed to connect with Baines' cut-back, but he made no such mistake in the 76th minute as substitute Fellaini turned past Williams in the box to give the striker a tap in.
That goal meant Everton had scored twice in a Premier League game for just the second time in 18 matches, and Vorm then made a fantastic scrambling save to deny substitute Denis Stracqualursi a third after more trickery from Pienaar