Emergency workers said that the bus carrying 56 people returning from France, most of them teenagers aged 14-16.
It appeared the death toll from the tragedy near Verona at around 11:00 pm (2200 GMT) could have been much worse had it not been for the bravery and quick thinking of a teacher.
Emergency workers said that the bus carrying 56 people returning from France, most of them teenagers aged 14-16 as well as several teachers and two drivers, ploughed into a bridge pillar.
A huge inferno very quickly engulfed the bus, dramatic pictures released by police showed, completely destroying the interior and reducing the vehicle to a burned-out wreck.
Some were killed when they were hurled from the coach on impact, but most of the victims were found badly charred inside the vehicle after being unable to escape in time.
The Hungarian government said that three people were in a very serious condition, including one in an artificial coma. Ten more were seriously hurt and 13 had minor injuries.
All were being treated in nearby hospitals. Twelve others who were not hurt were receiving counselling at a hotel in the vicinity.
"These were children. It's the hardest thing to take. Everyone thinks of their own," Hungarian Foreign Minister Peter Szijjarto, fighting back tears, told a news conference in Budapest.
"Of course the Hungarian government will do everything necessary to ease the families' pain."
Girolamo Lacquaniti, Verona traffic police chief, said on radio that no other vehicle was involved, pointing to mechanical failure or human error such as the driver falling asleep.
"The coach was travelling at quite a constant speed and we haven't found any traces of braking," Lacquaniti said. The vehicle burst into flames "moments after the impact," he said.
One of the drivers was among those killed, Italian media reported.
Describing a "devastating" scene, Lacquaniti told Radio Capitale that 13 passengers had managed to escape by jumping out.
"Those who were seated at the back of the bus saved themselves by smashing the windows amid shouting and panic," Hungary's consul in Italy Judit Timaffy told Italian media.
"A gym teacher rescued lots of those on board by getting back on the bus. He was taken to hospital with serious burns on his back," Timaffy said.
She said that the drivers had changed over around an hour before the accident.
Traffic police were still gathering evidence and consulting video evidence meaning it was "still too early to determine the cause of this tragedy," Lacquaniti said.
Once the blaze was put out emergency workers had to wait for a judge to give approval to comb through the smoking remains, which they then spent the rest of the night doing to establish the complete death toll.
At the Szinyei secondary school in the heart of Budapest, which is well known and has an excellent academic reputation, people were laying flowers and candles outside on Saturday, an AFP photographer said.
Hungarian Prime Minister Viktor Orban expressed his horror and shock.
"Losing children, young lives, is the worst thing for any family, community or nation," Orban said in a statement carried by state news agency MTI.
"At this time of mourning I am praying and am with the families and loved ones of those affected by this tragedy," Orban said.
Several parents of the children arrived at the Verona traffic police headquarters late morning on Saturday as well as 10 or so young people.