Three girls, two boys and a mother who had accompanied her child to the event, died in the incident. A police official told Sky Italia TV that the teenagers were all aged between 14 and 16, while the dead woman was 39.
More than 100 other people were injured, including 13 who were seriously hurt, officials said.
The local fire brigade said someone might have sprayed a substance like pepper spray into the crowd, triggering a panicked rush to the emergency exits.
La Repubblica newspaper quoted a survivor as saying that at least one of the exits was blocked. Another eyewitness said a wall inside the club had collapsed in the melee.
"People cannot die like this," said Deputy Prime Minister Matteo Salvini, who also serves as interior minister.
"We will find those responsible for these six broken lives, those who out of malice, stupidity or greed have turned a party night into a tragedy," he added in a statement. (Reporting By Philip Pullella and Crispian Balmer; Editing by Simon Cameron-Moore and Alexander Smith)
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