At least seven people were injured, with some in critical condition, local media reported.
The fourth-floor balcony collapse occurred during a birthday celebration at a building located two blocks from the University of California, Berkeley campus, local NBC affiliate KNTV reported.
The cause of the incident was not immediately known and an investigation was under way.
The Alameda County coroner's office confirmed that a total of six people had died.
Most of those involved in the accident were apparently Irish students on the J1 work-study visa program, which allows them to teach, study, conduct research and work legally during their stay in the United States.
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He said 13 people had been on the balcony at the time, citing police.
The structure collapsed shortly after the Berkeley Police received a phone call complaining about a noisy gathering.
"It is truly terrible to have such a serious and sad incident take place at the beginning of summer of adventure and opportunity for so many young people on J1 visas in the United States," Kenny said.
He said he had spoken with US ambassador to Ireland Kevin O'Malley, who presented the United States' condolences.
He told Irish national broadcaster RTE that "the police have indicated that they do not believe other nationalities to be involved."
Flanagan said consular officials in San Francisco were in contact with authorities in Berkeley and would provide assistance to those affected and their families.
A consular emergency response line was activated for those seeking information about loved ones.
Berkeley Police spokeswoman Jennifer Coats told AFP that officers received a call just before 1:00 am (0800 GMT).
Thousands of Irish students arrive in the US every summer on the J1 program. A J1 visa is seen as a "rite of passage" for many Irish students, with applications from Ireland amongst the highest in the world.