The hosts' chances of top-flight survival appeared to be fading until back-to-back wins against West Ham and West Brom offered cause for optimism.
Hopes of extending that run appeared to have suffered a blow when striker David Nugent pulled up with a calf injury in the warm-up on Saturday.
However, replacement Leonardo Ulloa answered the call by opening the scoring after 15 minutes, before Andy King pounced to double the lead with a minute left to play.
As a result, Leicester climbed to 18th – still inside the relegation zone but only on goal difference.
Swansea, meanwhile, looked like a team with little to play for and their wait for the point that will see them break their record Premier League tally of 47 goes on.
Visiting manager Garry Monk had more time than opposite number Nigel Pearson to consider his attacking options after Bafetimbi Gomis suffered a hamstring injury in last weekend's 1-1 draw with Everton. The Swansea boss opted to hand Nelson Oliveira just a second league start.
But, despite the late withdrawal of Nugent, it was the home side who started brightest and took a deserved lead in the 15th minute.
The Swansea defence failed to deal with a Marcin Wasilewski punt into the box and Ulloa was on hand to drive a shot into the bottom left-hand corner after being teed up by Wes Morgan.
Wayne Routledge chipped over goalkeeper Kasper Schmeichel into the Leicester net soon afterwards but the assistant referee’s flag had already been raised for offside, with television replays suggesting an extremely tight call.
Leicester continued to press, with Marc Albrighton curling a shot just wide, before Swansea finally began to show signs of life with 25 minutes on the clock – Jonjo Shelvey firing their first real chance high and wide.
Albrighton and Andrej Kramaric went close to extending Leicester's lead with early second-half efforts, while Shelvey continued to look Swansea's most potent threat at the other end, drawing a diving save from Schmeichel.
The Dane had to be alert as the second half progressed, denying Oliveira and and Shelvey in quick succession shortly after the hour mark as Swansea pressed for a leveller, while substitute Jefferson Montero's strike deflected wide a few moments later.
From the resulting corner, Leicester launched a rapid counter-attack, but Riyad Mahrez was unable to execute his attempted chipped finish, before Jamie Vardy saw claims for a penalty following a clumsy Ashley Williams challenge waved away.
Mahrez then forced Lukasz Fabianski to tip the ball around the post following a 20-yard strike late on, but King wrapped up the points when Fabianski dropped the ball into his path from Esteban Cambiasso's free-kick.
