A disciplined and destructive performance from Michael Laudrup's side gave the hosts an impressive four-goal lead to take to the Swedbank Stadion for the second leg next week
Early pressure from Swansea paid off after 36 minutes when Michu took advantage of a defensive error from Malmo's Filip Helander to give the hosts a much-deserved half-time lead.
A bright start in the second half saw newly-signed Wilfried Bony score twice in four minutes in his first competitive match for the club, and substitute and fellow new-boy Alejandro Pozuelo also found himself on the scoresheet as his half-volleyed shot beat Johan Dahlin in the Malmo goal to cap a fine team display ahead of next week's second leg tie.
Swansea boss Michael Laudrup was able to start recently signed Jordi Amat in central defence, whilst both Jonjo Shelvey and Bony were also handed their first competitive starts for the Swans.
Meanwhile, Malmo boss Rickard Norling was forced to make two changes to the side that lost 2-0 to Gefle in their last league game on Sunday, with Helander replacing Erik Johansson in defence, and Erik Friberg replacing Emil Forsberg, who dropped to the bench.
Swansea started the brighter of the two sides, with Shelvey playing a pivotal role. The former Liverpool man linked up with fellow new-boy Amat after only six minutes for Swansea's first real chance of the game, when the Spaniard met Shelvey's corner with his head, but Malmo defender Helander was placed perfectly on the line to clear the danger.
After a frantic start, Swansea's pace slowed down slightly and they reverted to the patient build-up play that brought them so much success last season. It paid dividends immediately, culminating in Michu's opening goal after 36 minutes, when he took advantage of an unfortunate slip from Helander and slotted the ball past Dahlin in the Malmo goal.
Swansea started the second half in more reserved fashion than the first, clearly concerned that Malmo's counterattacking play could cause them trouble if they pushed too far forward, but they continued to dominate possession and create chances in front of goal.
Malmo had an appeal for a penalty turned down early into the half when Magnus Eriksson seemed to be tripped over in the box by Angel Rangel, and Swansea went on to double their lead soon after, when Bony rose high to send a looping header into the top right-hand corner of the net.
It went from bad to worse for the Swedish side, as Bony made it 3-0 just four minutes later when he was in the right place at the right time to tap the ball in from close range after Michu saw his effort on goal rebound off the post.
The hosts weren't done just yet, however, and Pozuelo joined in on the fun when his half-volleyed shot left Dahlin motionless in the Malmo goalmouth and wrapped up a comprehensive and well-earned win for Swansea.