The attack occurred at about 0715 (0315 GMT) near the shrine of Maitham al-Tammar, whom Shia claim to be one of the early Shia sect followers, close to Kufa's grand mosque.
Most of the victims were Iranian visitors, Iraq's council of ministers said.
The attacker was driving a minivan behind two buses and detonated his explosives as people were getting off, police at the scene said . The buses were destroyed and a mosaic-lined wall of the shrine was damaged.
Nobody has claimed responsibility for the attack.
Asaad Abu Kallal, the governor of nearby Najaf, referring to members of Saddam Hussein's former party, said: "The purpose is clear - to stop Shia rituals. I suspect that the criminal Baathists are behind this act."
Kufa and its twin city Najaf, 160km (100 miles) south of Baghdad, are in a relatively peaceful area.
Sahib al-Amiri, an aide to radical cleric Muqtada al-Sadr said: "We denounce this act targeting innocent people and visitors. It was meant to shake the stability in Najaf."
The attack came as Nouri al-Maliki, the Iraqi prime minister, returned to Iraq after a regional tour to try to get support for his 24-point national reconciliation plan.
Al-Maliki has demanded an independent inquiry of the rape-slaying of an Iraqi girl and the killing of her family on Wednesday, as well as a review of the immunity rules protecting US forces from Iraqi prosecution.
Al-Maliki, whose brief tenure has been marked by several high-profile allegations of abuse by US forces, called for an Iraqi investigation - or at least a joint inquiry - into the March 12 rape-murder of Abir Qassim Hamza, and the killing of her mother, father and sister at their home in al-Mahmudiya, south of Baghdad.
Kufa and its twin city Najaf
are in a relatively peaceful area.
"We believe that the immunity given to members of coalition forces encouraged them to commit such crimes in cold blood (and) that makes it necessary to review it," he told reporters in Kuwait.
Former US soldier Steve Green was charged on Monday in a federal court in Charlotte, North Carolina, with rape and four counts of murder. He was held without bond. At least four other US soldiers still in Iraq are under investigation over the attack.
In Baghdad, an American military spokesman said that the US command was taking the allegations seriously and would discuss al-Maliki's demands.
The attack in Kufa comes after the US military predicted an increase in vehicle bombings, now that Abu Hamza al-Muhajir, also known as Abu Ayyub al-Masri, has succeeded the late Abu Musab al-Zarqawi as head of al-Qaeda in Iraq.
The US military reported 74 car bombs in the four weeks ending June 9, two days after al-Zarqawi was killed, compared with 125 in the four weeks since that date.