The country’s Interfax news agency quoted police as saying the assailants fled late on Monday after killing the imam of a mosque in the doorway of an apartment block where he lived.
A police officer was also wounded when trying to stop two armed people in the street.
Kislovodsk, like many parts of southern Russia, has a strong Russian nationalist Christian community which eyes with suspicion the growth of a Muslim population arriving from the impoverished Caucasus region.
Ethnic tensions in many parts of Russia have become a cause of concern for authorities, alarmed by the increase in racist attacks.
Analysts have said that by the year 2020, one in ten Russians could be Muslim.