Police said at least five people had been injured in fights between the two communities late on Monday.
"We have a responsibility to prevent the conflict in the region becoming a conflict in our society... in Germany," integration commissioner Annette Widmann-Mauz told the Funke newspaper group.
"I expect all sides, especially migrant organisations and religious communities, to take responsibility and contribute to restraint."
The commissioner advises the government on integration and serves as a point of contact for migrants and community organisations.
The clashes happened as around 350 people marched through the western city of Herne on Monday protesting Turkey's offensive in Kurdish-controlled areas of Syria, police said in a statement.
They were "provoked with hand signals" by people drinking at a nearby kiosk, the statement added.
"Some of them stormed into the kiosk, attacked two people inside and injured them" and they also broke a window.
The demonstration continued and someone threw a bottle at the marchers from a Turkish-owned cafe as they passed.
"The reaction was very emotional and angry" as several participants again rushed into the cafe, breaking windows and furniture and injuring at least one person inside, as well as a police officer who intervened.
Nevertheless, "the police were able to calm the situation".
Among the five people hurt was the organiser of the march. He, too, was attacked when he tried to stop the violence.
Of the roughly three million people with Turkish nationality or roots living in Germany, around one million are Kurds.
Politicians regularly warn of tensions between the two communities, which have been stoked by Turkey's offensive against Kurdish fighters inside Syria.
Call for calm
"According to the 30-year-old Turkish citizen, the men had assaulted him because of the jacket he was wearing, which had a Turkish national flag on it," a Berlin police statement said.
The leader of the Kurdish community in Germany, Ali Ertan Toprak, called Tuesday for calm.
"Our message is: do not let yourself be provoked. Do not react to provocations from the Turkish nationalist side," he told Funke.
"If there are riots, it will harm our casue," Toprak said. Kurds had "no interest in violence spreading on German streets".
Turkish troops moved last Wednesday into the Syrian border zone controlled by Kurdish militias, which helped a Western-led coalition fight Islamic State (IS) jihadists but are accused of terrorism by Ankara.
Germany, along with European allies such as France, has condemned the offensive and halted arms exports to Turkey.
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