In the day's other league game, bottom club Wigan Athletic were held 1-1 at home by West Bromwich Albion.
Sigurdsson, signed by Swansea on loan in January from German club Hoffenheim, followed up his double against Wigan this month by scoring in the 36th and 66th minutes as Brendan Rodgers's team clinched a third successive victory.
"This was our biggest game of the season purely because it came on the back of last week's win over Manchester City," Rodgers told Sky Sports.
"We don't want our players just to raise their game against the big teams and I thought they were absolutely phenomenal today. It's a very proud day for us all, players, fans and management team."
Wigan, however, were wasteful in front of goal throughout and their former midfielder Paul Scharner headed West Brom's equaliser midway through the second half.
Scharner's goal was the only effort on target all game from West Brom.
"In many ways we feel a sense of footballing injustice because we created chance after chance," said Wigan manager Roberto Martinez. "We could have won by three or four goals, that's how dominant we were.
"Unfortunately there have been too many times this season when our play has not been reflected in the scoreline. That's our big frustration."
Victory would have taken Martinez's team out of the relegation zone and they will look back ruefully on a series of missed scoring opportunites especially by Argentine forward Franco Di Santo.
