The incident took place in the Panjwai district of Kandahar province.
It is not clear what caused the explosion – the Taliban said they fired a rocket at it but police said there was no insurgent activity at the time.
The incident happened in the same area where a US soldier allegedly killed 17 Afghans in an attack on their homes.
Staff Sgt Robert Bales, 38, is accused of killing nine Afghan children and eight adults in Kandahar province on 11 March.
Security officials say that the fuel tanker was travelling from the city of Kandahar to Panjwai district at high speed when it overturned and caught fire, simultaneously setting a nearby civilian minivan ablaze.
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