A migration court in Malmö, southern Sweden, has overturned the agency's original decision, ruling that the three interpreters had a "special link" to Sweden, Swedish Radio News reports, citing local daily Sydsvenskan.
A fourth interpreter was denied residency because it was unclear whether he had served with Swedish forces.
Swedish troops used locally employed interpreters in Afghanistan between 2006 and 2014 to communicate with civilians and Afghan security forces.
Twenty-four interpreters made a joint asylum request at the Swedish Embassy mission at the Mazar-e-Sharif military base in July 2012, ahead of Swedish forces' withdrawal from the country.
A number of interpreters who have succeeded in travelling to Sweden have also been granted asylum, while several others have had their cases rejected.