Defender Gabriel Tamas, back in Romania after spells at Spartak Moscow and Auxerre, netted the only goal with 10 minutes to go with only the heroics of Sturm goalkeeper Christian Gratzei preventing Dario Bonetti's team winning by a greater margin.
Adrian Cristea was the first to threaten for the visitors with a left-foot shot from 25 yards which whistled over the top of Gratzei's crossbar.
Dinamo caused more distress in the Sturm box in the 13th minute when first Ferdinand Feldhofer blocked a shot from Marius Alexe before Djakaridja Kone's follow-up effort was charged down by Gordon Schildenfeld.
Alexe then brought a fine save from Gratzei, who tipped his 20-yard strike around the post before Schildenfeld cleared Adrian Cristea's corner.
Another corner, this time from Gabriel Torje, almost led to a goal but Ousmane N'Doye headed over from 12 yards.
Daniel Beichler was the first to threaten for Sturm after 20 minutes when he drew a save from Emilian Dolha and within a minute Peter Hlinka drilled an effort wide from 20 yards.
Gratzei pushed a shot from Torje round the post and then held on to another effort from the same player seconds later.
The Sturm goalkeeper was being kept busy as he then diverted an Andrei Cristea header wide from an Alexe cross.
The visitors were well on top and N'Doye's piledriver from more than 35 yards out forced Gratzei into more heroics. Two minutes before the break Samir Muratovic's free-kick drifted over the bar for the hosts.
Sturm were the first to go close in the second period through an Andreas Hoelzl shot which drifted wide and after 51 minutes Hlinka blasted a shot high over the top.
Beichler drilled an effort wide for Sturm before Gratzei was called on again to field a 30-yard drive from Tamas.
Zie Diabate shot over the top for Dinamo in the 73rd minute and less than a minute later Hoelzl's close-range effort had to be parried wide by Dolha.
Andrei Cristea was the latest to test Gratzei with 12 minutes to go and from the resulting corner from Claudiu Niculescu, Tamas applied the finish to score the only goal of the game.
Slovenian striker Klemen Lavric volleyed wide a Jakob Jantscher cross as time ran out in Graz.