The Germans clung on for three points and only won after Marko Arnautovic missed a great chance for the hosts in the 87th minute.
Reus burst down the left, cut inside Gyorgy Garics and buried a low shot past Robert Almer three minutes before the break after the feisty hosts, fielding nine Bundesliga-based players, had looked capable of producing an upset.
Ozil, scorer of two goals in the 3-0 win over Faroe Islands on Friday, converted a penalty after Veli Kavlak recklessly clattered into Thomas Mueller.
Austria, playing their opening match in Group C, refused to lie down and Zlatko Junuzovic pulled one back in the 57th minute, turning in the ball in after Arnautovic forced his way to the byline and sent a low cross into the goalmouth.
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