The Welsh club had failed to register a victory in the league since winning their first three games of the season - a run that stretched to five matches.
However, Bony found the net on two occasions at the Liberty Stadium to give Swansea an important three points ahead of a tricky run of fixtures that includes games against Everton, Arsenal and Manchester City.
The Ivorian's first capped a superb passing move after 34 minutes and was just reward for a first half in which Swansea totally dominated possession.
And Bony doubled the lead in the 57th minute for his fourth goal in his last three top-flight matches.
The result sees Swansea climb to sixth, level on points with Arsenal and Liverpool, while Leicester now sit 16th.
Jefferson Montero was selected to make his first Premier League start as Garry Monk made two alterations to the Swansea side that lost 2-1 at Stoke City last weekend, while Nigel Pearson - who bizarrely asked for the Liberty Stadium goals to be measured before kick-off - made four changes after a 1-0 loss at Newcastle United.
The hosts enjoyed the vast majority of the ball throughout the first half, with Bony getting the better of the early chances as he glanced a header wide of the left post before firing straight at goalkeeper Kasper Schmeichel from outside the penalty area after being teed up by Gylfi Sigurdsson.
The Icelander should have done better when Schmeichel fluffed a clearance in the 22nd minute, but Wes Morgan did just enough to get back and deny Sigurdsson the chance to shoot from close range.
Despite enjoying the lion's share of the ball, Swansea were struggling to create clear-cut chances, but a neat passing move unlocked the Leicester defence in the 34th minute.
After sending the ball to Sigurdsson with a delightful flick, Bony ran on to the return pass to put a clinical right-footed finish beyond the reach of Schmeichel.
Leicester failed to register a single shot - on target or otherwise - during the first half, but almost claimed a huge slice of luck when a long throw found its way onto the Swansea crossbar via a touch from Jonjo Shelvey three minutes into the second period.
Swansea have been guilty of not putting games to bed in recent weeks, and they will have been wary of letting another game slip as a brighter Leicester emerged after the break, with Danny Drinkwater stinging the palms of Lukasz Fabianski.
However, Monk's men made it 2-0 shortly before the hour mark when Montero evaded his marker down the left wing, subsequently squaring to Bony for a simple finish from a central position.
Shelvey rattled the crossbar with a sensational effort from 30 yards in the 63rd minute, but Riyad Mahrez gave home goalkeeper Fabianski plenty to think about with two decent efforts at the other end.
Leicester substitute Esteban Cambiasso should have reduced the deficit in the closing stages, but his effort hit the left-hand post after Fabianski had brilliantly parried Leonardo Ulloa's header, before the Polish stopper somehow scooped the ball to safety.
Full-back Angel Rangel, who reportedly cost just £10,000 when he arrived in south Wales from Terrassa in 2007, marked his 300th appearance for Swansea on Saturday, but it was £12million man Bony who stole the show as the hosts held on to end their winless run.
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