Michu opened the scoring in the first half before summer signing Bony netted twice after the interval on his competitive debut while Alejandro Pozuelo was also on target before the final whistle.
And manager Michael Laudrup also saw his side ensure Malmo, who hammered Hibernian in the previous round, failed to score any away goal as last season's Capital One Cup winners ended the game with a clean sheet at the Liberty Stadium.
This was the Welsh club's first foray into continental action since 1991, and they could hardly have wished for a better outcome as they took a stranglehold ahead of next week's second leg in Sweden.
It took them time to break down the Malmo rearguard, but when they did it came as little surprise that it was courtesy of Michu in the 37th minute when the Spaniard, scorer of 22 goals last term, produced a typically composed finish as he took advantage of a defensive slip.
Swansea survived what appeared a strong penalty appeal early in the second half when Jordi Amat made contact with Magnus Eriksson in the box but they responded by killing the game off.
Bony, signed from Vitesse Arnhem for £12million, claimed his first goal in a Swansea shirt by glancing a 55th minute header into the far corner from a Wayne Routledge cross.
The Ivory Coast international's second arrived five minutes later after Michu struck the post from Angel Rangel's pass, with Bony bundling in the rebound.
Pozuelo then rounded off a flowing move involving another new summer arrival, Jonjo Shelvey, Bony and Jonathan de Guzman with a volleyed finish late on.