One of those detained was the nephew of the Tunisian-born attack suspect Anis Amri, the country's interior ministry said.
The three men, aged between 18 and 27, were arrested yesterday and were members of a "terrorist cell... Connected to the terrorist Anis Amri"," it said in a statement.
It made no direct link between the trio and the attack on Monday, when Amri is believed to have hijacked a truck and used it to mow down holiday revellers at a Berlin Christmas market, killing 12 people.
The Berlin rampage was claimed by the Islamic State group, which released a video Friday in which Amri is shown pledging allegiance to IS chief Abu Bakr al-Baghdadi.
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The Tunisian interior ministry said Amri had sent money to his nephew and shared his jihadist views with him.
"One of the members of the cell is the son of the sister of the terrorist (Amri) and during the investigation he admitted that he was in contact with his uncle through (the messaging service) Telegram," it said.
The arrests come as German authorities are racing to find out whether Amri had help from accomplices before or after the attack.
"It is very important for us to determine whether there was a network of accomplices... In the preparation or the execution of the attack, or the flight of the suspect," federal prosecutor Peter Frank said Friday.
Seven of the victims were German nationals, a federal police spokeswoman told AFP. The other five came from the Czech Republic, Italy, Israel, Poland and Ukraine.
The fact that Amri was able to travel to Italy unhindered despite a Europe-wide arrest warrant has raised uncomfortable questions for intelligence agencies.
German security services have also faced criticism for not keeping better tabs on Amri before the Berlin carnage, even though he was a known criminal with links to the Islamist scene.
But Interior Minister Thomas de Maiziere denied there had been a blanket security failure.
It "is impossible to monitor every person suspected of posing a threat around the clock," he told the newspaper Bild am Sonntag.
Chancellor Angela Merkel has pledged a "comprehensive" analysis of how Amri was able to slip through the net and vowed to speed up the deportation of rejected asylum seekers like him.