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 on this Europa League third qualifying round tie ahead of next week's second leg in Sweden.
Michu opened the scoring in the 37th minute before record signing Bony grabbed a quickfire second-half double and fellow new recruit Alejandro Pozuelo rounded things off.
The Barclays Premier League side were in control of the contest throughout, with Jonjo Shelvey an eye-catching presence in midfield.
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.
The Spaniard, scorer of 22 goals last term, produced a typically composed finish as he took advantage of a defensive slip to slot the ball beyond the advancing Johan Dahlin.
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 Shelvey, Bony and Jonathan de Guzman with a volleyed finish late on.
Source: PA
Source: PA