The attack is a huge loss of face for the Milan-based firm which was set up in 2003.
The Corriere della Sera newspaper said the hackers posted internal documents, defaced the company's logo and made public embarrassing details of the firm's commercial dealings.
Company officials were unavailable for comment and its webpage was inaccessible.
Hacking Team sells the Da Vinci malware surveillance software to law enforcement agencies, according to media reports, although it claims to only deal with ethical governments.
It has been marked as an Enemy of the Internet by media rights watchdog Reporters Without Borders.
According to the leaked documents, the company's clients include outfits based in countries such as Sudan, Kazakhstan, Bahrain and Hungary.
Italian media said the leaked records showed the company had commercial transactions with companies and governments which used the malware to spy on journalists and political opponents.