Malmo travelled to the Czech Republic with one foot already in the League Phase, having secured a 3-0 home win in the first leg a week ago, and they almost killed the tie off in the opening minutes when Pontus Jansson's header hit the underside of the bar but stayed out.
But Sigma Olomouc showed they were up for the fight soon after, as Tomas Huk somehow put his header wide from three yards with goalkeeper Robin Olsen beaten, before a powerful Dominik Janosek effort from range curled wide.
Olsen proved his worth on 18 minutes with a decent save with his legs to deny Simion Michez, then at the other end 10 minutes on, Matus Hruska did the same to keep out Busanello's effort. Sead Haksabanovic followed up well on the rebound, but his strike was decisively blocked on the line.
Haksabanovic scored twice in the reverse fixture and he continued to look likely to break the deadlock again in the second half, shrugging off challenges before firing a whisker wide just before the hour mark.
But soon after, MFF did take the lead after a corner was played short to Taha Ali, who tried to catch Hruska out at his near post and though the goalkeeper diverted the effort onto the woodwork, Daniel Tristan Gudjohnsen was perfectly placed to stab home the rebound.
As the half wore on, Daniel Vasulin blazed the ball over from a promising position, though the home side were already in consolation territory in terms of the tie as a whole. Gudjohnsen almost grabbed himself a second but Hruska got down well to thwart his effort.
The visitors did put the icing on the cake in the first minute of added time - a through ball from Lasse Johnsen cut the Olomouc defence in two and Emmanuel Ekong raced through to poke the ball past the onrushing Hruska.
The 2-0 win on the night and 5-0 aggregate success sees Sweden represented in the UEL League Phase, the draw for which will take place tomorrow. MFF will be hoping for better than the five points they mustered from eight matches at the same stage a season ago.