It took until the 17th minute of their third game for Malmo's first goal of the group stage to arrive, but when it did the Swedbank Stadium erupted as Rosenberg confidently stabbed home Pa Konate's low cross from around the penalty spot.
Jo-Inge Berget almost doubled the home side's lead just before halftime but Andriy Pyatov pulled of a wonderful reflex save to push the ball away.
Shaktar captain Darijo Srna conceded a penalty 10 minutes into the second half when he pulled down Jo-Inge Berget, but Nicola Djurdjic missed the spot-kick.
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Shaktar, who looked dangerous going forward but had few shots on target, managed to get the ball in the net in the 78th minute but the goal was ruled out for offside.
The win left Shaktar firmly rooted to the bottom of Group A without a point, while Malmo are in third place with three.
Paris Saint Germain and Real Madrid are level at the top on seven points after a 0-0 draw in the French capital.
(Editing by Ed Osmond)
