Malmo conceded what proved to be the decisive goal with just 10 minutes left to play, when Martin Ericsson slotted home a finish between the legs of stand-in keeper Andreas Vindheim after poor defending at a corner-kick.
Top striker Rosenberg looked lively throughout, having missed the first leg against Celtic through suspension, but passed up a number of opportunities.
The Swedish international will start the game against Ronny Deila’s men, but, on this showing, he will need to sharpen his finishing touch in front of goal if he is to trouble the Celtic defence.
The defeat leaves Hareide’s side in fifth place, four points adrift of Swedish Allsvenskan leaders IFK Gothenburg, having also played a game more.