The blast on Wednesday comes less than a week after 11 Chinese road workers were shot dead near the city.
Provincial police chief Mutalib Bek said the remote-controlled device exploded at about 05:00 GMT on a busy road as a vehicle used by Kunduz-based German peacekeepers passed by.
A fifth person was seriously wounded in the attack.
Kunduz governor Muhammad Umar said the driver, two children and an elderly man were “martyred”.
He was not sure whether the driver was killed by the blast or as a result of crashing into a wall shortly afterwards.
The bomb was probably targeting a provincial reconstruction team (PRT) made up of some 250 German soldiers.
“The mine exploded as the PRT vehicle arrived on the spot,” Omar said.
Several PRTs have been established in Afghan provinces to extend the reach of the central government, improve security and aid reconstruction.
On Thursday, 11 Chinese road workers were killed as they slept in tents pitched south of Kunduz city.
The attack, in which gunmen riddled the tents with machinegun fire, was the worst against foreigners in Afghanistan since the fall of the Taliban in late 2001.
So far 14 Afghans have been arrested in relation to the attack.