Police said the target of the bomber appeared to have been a white pickup truck similar to the kind often used by security contractors in Afghanistan. The truck was partially destroyed in the blast.
The head of Kabul police, Abdul Rahman Rahimi, said one person had been killed and 13 wounded, including three women. He said the aim of the attack in an overwhelmingly civilian area had been to create fear among Afghans.
"Enemies of humanity detonated a suicide car bomb in front of a madrassa where children were learning the Koran and Islamic studies. It shows that they are enemies of mosques, God and the Koran," he said.
A minibus was also destroyed by the explosion, which shattered the windows of nearby shops and spread debris across the street.
The attack, the latest in a recent series of suicide bombings, came a day after the powerful head of Pakistan's army, General Raheel Sharif, visited Kabul for talks intended to lay the groundwork for a resumption of peace talks with the Taliban.
The Afghan Taliban's main spokesman Zabihullah Mujahid said on Twitter the suicide attack had targeted a convoy of foreign forces and had caused heavy casualties.
The Taliban often exaggerates casualty tolls in attacks on Afghan and foreign forces.
A NATO spokesman in Kabul said there was no indication that any member of the its mission had been hit.
Taliban insurgents have claimed a number of attacks this month, including an assault on a Spanish embassy guesthouse in the capital that began on Dec. 11 and a suicide bombing near Bagram air base that killed six American troops last week.
They have also been involved in heavy fighting in their historic heartland in Helmand province in southern Afghanistan, where they have been battling government forces for weeks for control of Sangin district.
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.