The on-loan midfielder played a major role as Brendan Rogers' side registered only their second Premier League away win of the season.
The Iceland international, who joined from Hoffenheim last month until the end of the season, cancelled out Marc-Antoine Fortune's opener and laid on the winner for Danny Graham.
The visitors dominated for long spells as they followed up their 1-1 home draw against Chelsea in mid-week with another impressive performance.
But Albion's woeful home record continued and they have collected just eight points at The Hawthorns compared to 18 on their travels.
Roy Hodgson's side were often left chasing shadows with Sigurdsson and Joe Allen pulling the strings in midfield and 10-goal Graham a constant threat.
Swansea were soon into their stride and created the first opportunity of the game.
Sigurdsson's shot was blocked from an Allen corner and the follow-up attempt from Angel Rangel was deflected just wide.
Baggies midfielder Youssouf Mulumbu was caught in possession and was relieved to see Allen drag his shot past the post.
James Morrison brought Sigurdsson down 25 yards out and the Swansea player took the free-kick himself and sent it just wide.
In a rare West Brom attack Gareth McAuley headed wide from a Nicky Shorey cross. Peter Odemwingie's control then let him down when he was found in space by Jerome Thomas.
The threatening Graham missed a good chance to put Swansea ahead after half an hour. Graham shrugged off the challenge of McAuley when challenging for a Leon Britton free-kick. But, from 12 yards out he shot wide when he should have at least tested Ben Foster.
Albion enjoyed their best spell approaching half-time with Sigurdsson blocking a goal-bound flick from Jonas Olsson.
Michel Vorm was alert to hold onto an Olsson flick from a Graham Dorrans free-kick.
Britton was booked for bringing down Thomas and Swansea appealed in vain for a penalty after a challenge by McAuley on Scott Sinclair.
Olsson was looking Albion's most threatening player at set pieces and his header was blocked from a Dorrans corner.
But it was against the run of play when Fortune put West Brom ahead after 53 minutes.
A Dorrans corner was flicked on by Olsson and at the far post Fortune had time to steady himself before drilling the ball into the corner of the net.
Swansea needed only two minutes to draw level through Sigurdsson.
Britton's ball released the over-lapping Neil Taylor and his low cross was side-footed past Foster by Sigurdsson.
The game turned around with Graham putting Swansea in front after 59 minutes.
This time Sigurdsson was the creator with a low ball to the near post and Graham made no mistake for his 10th goal of the season.
Odemwingie was guilty of a glaring miss when he shot over with the goal at his mercy after a mis-hit effort from Dorrans had landed in his path.
Then Paul Scharner failed to make proper contact on his shot from 10 yards out after Somen Tchoyi had pulled the ball back into his path.
Morrison was booked for a foul on Britton as Swansea continued to look the more likely to score.
Olsson blocked a shot from the impressive Sigurdsson but in injury-time Vorm denied Fortune to prevent an undeserved Albion equaliser.