The history of the museum begins with the Kuban Cossack Ethnographic and Natural History Museum, founded in 1879 under the Kuban Regional Statistical Committee. Its creator was the regional historian and Cossack senior officer Yevgeny Dmitrievich Felitsin (1848-1903). His contemporaries called him the "encyclopedia of the Caucasus" and a "living chronicle," as he had an excellent knowledge of the history and geography of the region, as well as archaeology, ethnography, cartography, geology, botany, and mineralogy.

The first exhibits of the museum were gifts and donations from the public, as well as Felitsin's personal collections on archaeology and ethnography. To acquire them, he used his modest salary and often went into debt. Thanks to him, the museum possesses one of the best collections of stone "Polovtsian" sculptures in the world, gathered from across the Kuban region. By the first year, the collection numbered 276 archaeological and 42 ethnographic items.

Felitsin enthusiastically studied the ancient history of the peoples of the North Caucasus, identifying and collecting archaeological monuments. In 1879, he discovered two Greek slabs in Yekaterinodar, which were considered lost, and sent them to Moscow. The slabs bore the names of Levkon, the king of the Sind, and Ksenoklid (4th century BC). Felitsin also created a unique archaeological map of the antiquities of the Kuban region, published in 1882 by the Imperial Moscow Archaeological Society.

Yevgeny Dmitrievich paid special attention to monuments of Russian history. In the autumn of 1879, he visited Taman, where he researched the stay of M.Yu. Lermontov. Felitsin made drawings and described the Cossack's hut of Mysnik, where the poet stayed. These studies helped create the historical and ethnographic exhibition of the Lermontov Museum in Taman in 1976.

In March 1880, Emperor Alexander II awarded Felitsin the Order of Stanislaus III degree for his work. He was elected an honorary member of various statistical committees and societies, including the Moscow Archaeological Society and the Imperial Society of Naturalists, Anthropology, and Ethnography.

After Felitsin's death, his library, documents, and photographs were dispersed among private collections. Only in 1909, six years after his death, did the appointed ataman Mikhail Babych issue a decree to collect Felitsin's personal archive and transfer it to the Cossack Museum.