Officials said the camp in Malakasa, some 38 kilometres (24 miles) northeast of Athens, was placed under “full sanitary isolation” for 14 days, with no one allowed to enter or leave.
A migration ministry source said the Malakasa facility currently shelters over 1,700 people.
The ministry said the 53-year-old Afghan man had sought help with virus symptoms at the in-camp medical facility.
He was subsequently taken to an Athens hospital where he tested positive, and his family was quarantined.
A screening of the camp is in process, the ministry said. The ministry said this would involve staff, close contacts of the ailing man and any with suspect symptoms.
The health ministry has ruled out a general screening campaign at the camps, noting that such a process would be costly and impractical.
The first camp outbreak surfaced earlier this week at a facility in Ritsona near Athens where 23 people – out of more than 2,500 – have tested positive so far.
In Greek migrant camps, where tens of thousands of asylum seekers live in dire conditions, regulations had already been announced in March to keep residents in all camps as far from the local population as possible.
Migration Minister Notis Mitarachi has warned that asylum seekers who attempt to break the lockdown will be prosecuted.
Greece is sheltering some 100,000 asylum seekers, mostly in camps and in hotel rooms and flats.
Social distancing is practically impossible as thousands of refugees and migrants jostle in the queues for toilets and showers in the camps.
Greece, a country of 11 million, thus far has registered 73 deaths and 1,735 cases.