The students were enrolled at Barcelona University as part of the European Erasmus exchange programme, said Jordi Jane who heads interior matters for the Catalonia region.
The bus was carrying 57 people in all, including the driver. 30 were rushed to hospital after the accident took place on Sunday, officials added.
13 people – two Germans, seven Italians, one French student, one Austrian, one Romanian and one from Uzbekistan – were confirmed dead while six of the injured remained a critical condition on Monday.
The student passengers were aged between 20 and 32 and came from some 20 countries including Sweden, Norway, the UK, Switzerland, Japan and New Zealand.
They were all aged between 19 and 25 and were enrolled at Barcelona University as part of the Erasmus student exchange programme.
The crash occurred just before 6am on Sunday near the small town of Freginals, about 150 kilometres (95 miles) south of Barcelona as the bus was returning from a traditional festival in the eastern city of Valencia.
The driver "hit the railing on the right and swerved to the left so violently that the bus veered onto the other side of the highway," Jane said.
The bus then hit a car coming in the opposite direction, injuring two people inside, he added.
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Spanish television showed images of the car, the front of which was smashed in, and the bus lying on its side after the accident.
Catalonia's newly-elected president Carles Puigdemont was due to travel to Paris on Sunday but cancelled the trip to go to the site of the accident.
The region's high court said in a statement that an initial probe revealed "the bus driver tested negative for drugs and alcohol".