Some 30 refugees took up home in the block in Köpenick, eastern Berlin, at the weekend, with more expected to follow shortly.
The controversial blocks are being erected as an emergency measure following an influx of refugees both to the capital and to Germany during 2014, but have seen a spate of protests by locals, neo-Nazis and counter-demonstrations by anti-fascist groups.
Up to 400 asylum-seekers will be housed in the Köpenick block, which was thrown up in just five weeks.
The brightly-coloured block is the first of six planned "container towns" around the capital, with further ones coming onstream in Pankow, Lichtenberg, Marzahn and Lichterfelde. Rooms are divided into twin-beds, or family rooms, with communal kitchens.
Both the Marzahn and Köpenick blocks have seen protests by neo-Nazis and right-wing activists before they were even finished. Most of the planned blocks are in deprived areas of the capital.
The Marzahn block also had its Christmas tree stolen at the weekend, with police suspecting neo-Nazis of being behind the theft, according to BZ newspaper.
Berlin received some 1,471 refugees in December so far, taking the total for 2014 to around 15,000.
Across Germany as a whole some 200,000 refugees arrived in 2014, double the figure for 2013. That figure expected to increase in 2015 to around 230,000, acrcording to Manfred Schmidt, head of the Federal Office for Migration and Refugees earlier this month.
Those number form the backdrop of anti-Muslim feeling stoked by the up to 17,000-strong Pegida (Patriotic Europeans against the Islamization of Europe) rallies being held weekly in Dresden, with smaller offshoots in other cities.
In Berlin a long-running stand-off continues between some refugees, estimated at 30-40, squatting in an abandoned school in Kreuzberg, and the riot police who attempted an eviction in the summer.
The Pegida protests have pushed Germany's relatively liberal asylum policies up the political agenda, with Chancellor Angela Merkel appealing against "agitation and mud-slinging" and president Joachim Gauck using his traditional Christmas address to appeal for tolerance and understanding.
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