In an email to the Al-Arabiya station, the group Jundallah (Soldiers of God) said Thursday's attacks were a response to the recent execution of one of its leaders and threatened more such attacks.
The blasts targeted Shia Muslim worshippers and Revolutionary Guards members in the predominantly Sunni city of Zahedan in Sistan-Baluchestan province, the Fars news agency quoted Iran's deputy interior minister as saying.
"Hours ago, a suicide operation was carried out in the Jamia mosque, which left several martyrs and several wounded," the agency quoted Ali Abdollahi as saying.
"The first explosion took place behind a checkpoint and a number of Revolutionary Guard members were killed and injured because of it."
The attack came as people celebrated the birthday of Imam Hussein, grandson of the Prophet Mohammed, a day also set apart each year to honour the Revolutionary Guards.
Jundallah says it is fighting for the rights of the Sunni Baluch minority, and accuses Iran's Shia-dominated government of persecution.
His younger brother, Abdulhamid, was executed in May in Iran after being captured in Pakistan in 2008 and extradited to Iran.
Hossein Ali Shahriari, a Zahedan member of parliament, told Fars that there were two suicide attacks one after another, with the first one carried out by a bomber dressed as a woman.
"The attacker, dressed in women's clothing, was trying to get in the mosque, but was prevented" when the first blast took place, Shahriari said.
"When people came to rescue those hit in that blast, another bomber blew himself up. Three to four have been killed at least in the first attack."
Jundallah has previously carried out several deadly attacks in Sistan-Baluchestan- including on the elite Revolutionary Guardscorps.
The United States described the attacks as "horrific", and called for the perpetrators to be held accountable.
"I condemn in the strongest possible terms today's terrorist attacks claimed by Jundallah that targeted Iranians at a mosque in the Sistan-Baluchestan province of Iran," Hillary Clinton, the secretay of state, said in a statement.
"The United States extends its sympathy to the families and loved ones of those injured and killed.
"We also call for the perpetrators of this horrific attack to be held accountable for their actions.
"This attack, along with the recent attacks in Uganda, Pakistan, Afghanistan, Iraq, and Algeria, underscores the global community's need to work together to combat terrorist organisations that threaten the lives of innocent civilians all around the world."
Updates – Malaysia Airlines Flight 370 goes missing
14 hours ago