At least 38 people were killed in the blast, said Abubakar Zakari Adamu, a spokesman for the Kaduna state Emergency Management Agency. Others suffered serious injuries and were receiving treatment at local hospitals, Adamu said.
A witness, Augustine Vincent, said he was riding a motorcycle just behind the car when it exploded.
"God saw our heart and saved us," he said.
Churches have been increasingly targeted by violence on holy days in Nigeria, a nation of more than 160 million people of Christians and Muslims. A Christmas Day suicide bombing of a Catholic church in Madalla near Nigeria's capital killed at least 44 people.
Police and soldiers quickly cordoned off the blast site, though citizens looked on at the flames and damage. Authorities said they had no immediately suspects in the attack, though a radical Islamist sect known as Boko Haram has claimed similar attacks in the past.
Boko Haram, whose name means "Western education is sacrilege" in the Hausa language of Nigeria's north, is waging an increasingly bloody fight with security agencies and the public. More than 380 people have been killed in violence blamed on the sect this year alone, according to an Associated Press count.
The sect, employing suicide bombers and assault-rifle shootouts, has attacked both Christians and Muslims, as well as the United Nations' headquarters in Nigeria.