They were in "a life-threatening situation," the ministry said, adding that the 30-year-old Romanian driver tried to flee but was later caught by a police sniffer dog.
It comes less than one week after 71 bodies were found in an abandoned truck on a highway a few miles further south, near the Slovakian border.
Vienna police spokesman Thomas Keiblinger said: "At first [officers] were not able to open the vehicle but heard people`s voices coming from the back of the van. They then broke open the doors and found 24 people penned in the back. The majority of these young people were 16 or 17 years old."
He added that the space in the back measured 6ft 6in by 9ft 10in and was only 5ft 10in high.
"The police really came across a terrible situation," Keiblinger said. "The scene there was really awful. The people were sitting on top of each other, standing on top of each other and had no chance of opening the door themselves from inside.
He added: "The doors were welded shut, the windows had bars over them. These people were not able to free themselves."
Interior Minister Mag Johanna Mikl-Leitner said: "This arrest is for me a sign that our actions against the traffickers have an effect."
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