Swansea's third win in their last four league matches has moved the Welsh outfit above Leicester in the table, with five straight defeats now leaving Claudio Ranieri's side just one point outside of the bottom three in 17th position.
As for Leicester, Demarai Gray was handed just his fifth Premier League start of the season, while Marc Albrighton also returned to the XI as Ahmed Musa and Shinji Okazaki both dropped to the bench for the English champions.
Swansea started on the front foot and had a decent penalty shout in the third minute when Leicester captain Wes Morgan looked to handle a strike from Sigurdsson, but the referee was not interested and waved play on.
It took until the 33rd minute for either side to create a serious opportunity and it was Routledge that struck a volley towards the bottom corner, but Leicester goalkeeper Kasper Schmeichel was on hand to make a smart save.
For all the firepower on the field, it was a defender that scored the game's opening goal in the 36th minute when Swansea centre-back Alfie Mawson hammered a sensational volley past Schmeichel after Federico Fernandez had teed up his teammate with a clever header.
Swansea were not finished in the first period, however, and doubled their lead in the second minute of stoppage time when Sigurdsson found Martin Olsson with a clever pass and the left-back thumped one past the stranded Schmeichel, who was again left exposed by his static defence.
Ranieri introduced both Islam Slimani and Ben Chilwell at the interval, but it was more of the same at the start of the second period as Swansea continued to look the more likely to add to the scoring in Wales.
Sigurdsson lined one up in the 54th minute as Paul Clement's side looked for a third, but Schmeichel was on hand to make another smart save as Leicester just about remained in the contest approaching the final 30 minutes of action.
Slimani had a golden chance to halve the deficit in the 63rd minute after Mahrez released the centre-forward inside the Swansea box, but Lukasz Fabianski was on hand to save, before Wilfred Ndidi missed the target with a curling effort moments later.
Ayew was introduced for his Swansea debut in the 72nd minute during a period where Leicester were looking dangerous, but the majority of the away side's attempts came from distance as they struggled to break the lines.
Vardy had a decent chance to get Leicester on the scoresheet 10 minutes from time after a smart pass from Slimani found the Englishman in space, but his effort was wide of the post as Swansea survived another dangerous moment.
Leicester continued to throw players forward in the latter stages in search of a goal, but Swansea held strong to claim their first clean sheet under Clement and put another important three points on the board.
Swansea now have a two-week break until their next Premier League match away to Chelsea on February 25. Leicester, however, are in FA Cup fifth-round action away to Millwall next weekend, before travelling to Sevilla on February 22 for the first leg of their last-16 clash in the Champions League.
The struggling Foxes do not return to league action until they host Liverpool on February 27.