The van with 11 passengers approached a patrol at high speed in the province of Helmand on Wednesday and did not heed hand signals stop, British spokesman Lieutenant Colonel Andy Price told AFP.
Initial investigations said a warning shot fired into the ground ricocheted and hit the vehicle, killing two adults, he said.
"It was definitely fired as a warning shot and we are deeply upset we have lost two lives. There was no intention to cause harm to the local civilians," Price said.
The NATO-led International Security Assistance Force, under which the nearly 5,000 British troops in Afghanistan serve, said the loss of life and injury was regretted and an investigation had begun.
The deaths come at a sensitive time for the 37-nation ISAF after troops were accused of killing scores of civilians in an anti-Taliban raid in the adjoining province of Kandahar on October 24.
The force has yet to release the findings of its inquiry into the incident but The New York Times reported this week that a joint NATO and Afghan investigation found 31 civilians were killed in the night-time air attack.
Many were nomadic shepherds who had fled their tents with their wives and children after a NATO bomb struck a nearby compound, killing 20 Taliban fighters, the paper cited a NATO official saying on condition of anonymity.
The attack was in the Panjwayi area, about 35 kilometres (19 miles) west of Kandahar city, and ISAF said at the time it had been aimed at Taliban rebels who were re-infiltrating after a defeat in September.
The September offensive killed about 1,000 Taliban, NATO has said, while a government enquiry found 53 civilians also died.
Human rights groups have raised concern about the mounting civilian toll in the battle between Taliban-led insurgents and the security forces.
Around 1,000 civilians have been killed in the violence this year, according to Human Rights Watch.

Copyright AFP (Agence France-Presse), 2006