Cage has campaigned since 2003 on behalf of people who have allegedly suffered abuses in the United States' fight against terrorism.
Rabbani told the court he wanted to protect a Qatari client.
"It was a case involving the US against an individual who was allegedly tortured over the course of 12 or 13 years in US custody," he said.
"There were around 30,000 (documents) which I was especially uncomfortable handling and I felt an enormous responsibility to try and discharge the trust that was given to me," he said.
But the court dismissed Rabbani's arguments and handed him a 12-month conditional discharge, meaning no further action would be taken unless he commits a further offence.
Rabbani, 36, was also ordered to pay a fine of £620 ($835, 705 euros) and police have seized his phone and laptop.
Speaking outside court, Rabbani said the law amounted to a "digital strip search" and called for the legislation to be changed.
"I took the decision not to raise the details of an important torture case before my arrest and ultimately I have been convicted of protecting the confidentiality of my client," he said.
"If privacy and confidentiality are crimes, then the law stands condemned."
The head of London's counter-terrorism command, Dean Haydon, said the law under which Rabbani was convicted was crucial "to help keep the public safe".
"Schedule 7 of the Terrorism Act 2000 is a vital tool in the fight against terrorism and we are committed to ensuring the power is used appropriately and proportionately, as it was in this case," he said in a statement.
Schedule 7 was also used in 2013 against the partner of Glenn Greenwald, the US journalist behind the Edward Snowden intelligence leaks.
David Miranda, a Brazilian citizen, was held for nine hours at Heathrow airport while police questioned him and seized items including his phone and laptop.