Ban was speaking at a meeting with British international development minister Andrew Mitchell, according to a statement from Ban's spokesman.
"The Secretary General said he was shocked and outraged to hear of the incident in Tehran in which demonstrators entered the British embassy, briefly abducted embassy staff and damaged property," the statement said.
Ban welcomed the Security Council's swift statement condemning the incident and called on Iranian authorities to investigate the incident and avoid any repetition.
Protesters in Tehran trashed British offices and stole documents in attacks that dramatically escalated the West's standoff with Iran over its nuclear programme.
US Secretary of State Hillary Clinton also demanded that Iran protect foreign diplomats and strongly condemned the storming of the British embassy.