Unknown attackers hurled incendiary devices at the centre in the town of Ahlen, police said, also telling national news agency DPA they were not ruling out a political motive.
At the weekend molotov cocktails were thrown at Turkish community mosques in Berlin and the town of Lauffen, a cultural centre in Meschede and a Turkish vegetable shop in Itzehoe, where the windows of a mosque were also smashed.
No-one was wounded in the attacks.
Police arrested three Syrian men, who denied involvement, after the Meschede attack, DPA reported.
The attacks came on a weekend that saw several demonstrations against Turkey's military operation to oust the Kurdish People's Protection Units (YPG) from the Afrin region of northern Syria.
Pro-Kurdish demonstrators rallied in Hamburg Saturday, where rocks were thrown at the Turkish consulate, and on Sunday at Düsseldorf airport some protesters scuffled with police who deployed pepper spray.
Turkey's foreign ministry voiced "deep concern" about the attacks and said on Sunday that the incident in Lauffen had been claimed by a group affiliated with the banned Kurdistan Workers Party (PKK).
It added that there had been "a considerable increase in the number of attacks against Turkish mosques in Germany recently by racist and Islamophobic groups as well as the PKK terrorist organisation".
"We expect the perpetrators of these attacks to be caught as soon as possible and brought to justice and necessary measures to be taken by the German authorities to prevent the recurrence of such attacks."
A pro-Kurdish website has published purported videos of the Lauffen and Meschede attacks, claiming they were carried out by Kurdish youths.
The Kurdish Community in Germany group sharply condemned the attacks as well as social media calls that were "urging Kurdish youths to employ violence against Turkish institutions".
Its chairman Ali Ertan Toprak said such attacks "endanger innocent human lives and politically damage primarily the cause of the Kurds while jeopardising peaceful coexistence in Germany".
Toprak urged Kurds to refrain from violence, stressing that it was unclear whether the attacks were carried out by "supporters of the PKK or the Turkish secret service MIT".