Looking the make the most of their home advantage, Dinamo had to wait just four minutes before they took the lead through Sammir.
A neat pass by Ante Rukavina found Jerko Leko racing down the right, and while his cross was deflected away, it fell kindly to Sammir who slotted home clinically.
Buoyed by their goal, Dinamo continued to plough forward, but it was the visitors who equalised against the run of play, after Agon Mehmeti raced onto Markus Halsti's slide-rule pass and cut inside Luis Ibanez, before beating Ivan Kelava at his near post.
Despite all of Sammir's efforts, Malmo held on till halftime and would have been forgiven for being optimistic in their chances of holding out for a draw.
However, all that changed eleven minutes into the second period, when Ibanez advanced from his defensive position to deliver a perfect cross for Rukavina to touch home at the near post.
Just four minutes later, Sammir added another for the Croatian outfit, after he was felled in the box by Malmo goalkeeper Dusan Melicharek.
And six minutes from the end, Dinamo capped off a wonderful night and a sizeable advantage for next Tuesday's return leg when Sammir played in substitute Fatos Beciraj, who kept his cool to seal a 4-1 win for the Croatian side.