“The target was Ismail Khan,” Zabihullah Mujahid, a Taliban spokesman who claimed responsibility for the attack, told the Reuters news agency by telephone from an undisclosed location.
Khan, a former governor of Herat, a province bordering Iran, is a senior member of the Northern Alliance whose forces helped US forces in toppling the Taliban in 2001.
The attack wounded the minister’s bodyguards, who are being treated in hospital for minor injuries, said Naqibullah Arwin, a spokesman for the local governor’s office.
Witnesses told an AFP correspondent at the scene that the blast was set off by a suicide car bomber.
Afghanistan has been reeling from renewed Taliban attacks, which have stalled a Western-backed reconstruction drive.
Thousands of people have been killed and Nato and the US – with more than 100,000 troops in Afghanistan – remain bogged down in a nearly eight-year conflict.
The violence is at its deadliest level in the eight years since the 2001 US-led invasion ousted the Taliban regime in Kabul.