The reigning Swedish champions had a dream start to the match and Joseph Ceesay scored 1-0 in the sixth minute after a brilliant strike.
The home team equalized not long later, but after just under an hour, MFF tightened the grip again when Isaac Kiese Thelin made it 2-1 with a well-placed finish at the goalkeeper's left post.
But the joy was short-lived.
In the 69th minute, team captain Teddy Teuma scored the equalizer from distance - and Union SG got energized.
Just two minutes later, the winning goal, 3-2, came from Victor Boniface.
Malmö are now last in Group D with two straight losses, after losing 0-2 at home to Braga last week.
The text is updated.