The new government under conservative Chancellor Friedrich Merz has frozen the programme, but Berlin's administrative court said the family had been given a "legally binding" commitment and must be allowed to travel to Germany.
About 2,500 Afghans with similar approvals are estimated to be in Pakistan waiting to be able to travel to Germany.
The programme was set up for those who worked with Germany's army or other institutions in Afghanistan, as well as journalists and activists judged to be threatened by the Taliban.
However, the new government under Merz -- who made a crackdown on immigration one of his key election pledges -- has said in its coalition agreement it wants to end the programme "as far as possible".
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The family concerned by Tuesday's decision is currently in Pakistan and had lodged an urgent appeal, telling the court they were about to be deported to Afghanistan where they would be in fear of their lives.
The family was given approval to come to Germany in 2023 under the programme.
They then applied to the German embassy in Islamabad for visas but these have not been issued.
The court said the government had the right "to decide whether and under what conditions the admission programme for Afghan nationals will be continued" and could "abstain from new approvals during this decision process".
But "it cannot release itself" from approvals which have already been given, it said.
According to the last update given about the scheme given in April 2024, more than 33,000 Afghans have travelled to Germany under the programme.
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